UNIVERSITY OF CALIFORNIA,IRVINE

Measurability Properties on Small Cardinals

DISSERTATION

submitted in partial satisfaction of the requirementsfor the degree of

DOCTOR OF PHILOSOPHY

in Mathematics

by

Monroe Blake Eskew

Dissertation Committee:Professor Martin Zeman, Chair

Professor Itay NeemanProfessor Svetlana Jitomirskaya

2014

c© 2014 Monroe Blake Eskew

DEDICATION

To Courtney

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS v

CURRICULUM VITAE vi

ABSTRACT OF THE DISSERTATION viii

Introduction 1

1 Preliminaries 61.1 Basic combinatorics of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Forcing with ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Elementary embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Dense ideals from large cardinals 172.1 Layering and absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 The anonymous collapse . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 An unfortunate reality . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Construction of a dense ideal . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Minimal generic supercompactness . . . . . . . . . . . . . . . . . . . 332.2.2 Dense ideals on successive cardinals? . . . . . . . . . . . . . . . . . . 35

3 Structural constraints 373.1 Cardinal arithmetic and ideal structure . . . . . . . . . . . . . . . . . . . . . 393.2 Stationary reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Nonregular ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Ulam’s problem and regularity of ideals 484.1 Generalizing Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Reduction to normality and degrees of regularity . . . . . . . . . . . . . . . . 54

5 Consistency results from generic large cardinals 595.1 Foreman’s Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Preservation and destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Compatibility with square . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4 Mutual inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

iii

6 Coherent forests 786.1 Aronszajn forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Influence of the P-ideal dichotomy . . . . . . . . . . . . . . . . . . . . . . . . 866.3 Suslin forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography 100

iv

ACKNOWLEDGMENTS

The completion of my PhD was aided in part by GAANN funding from the US Departmentof Education during the academic years 2008-2009 and 2009-2010. I am also grateful to theUCI Mathematics Department for financial support through several Teaching Assistantshipsand Departmental Fellowships.

I am indebted to Martin Zeman and Matthew Foreman, who taught me set theory.

v

CURRICULUM VITAE

Monroe Blake Eskew

EDUCATION

• University of California, Irvine

– MS Mathematics, 2009.– PhD Mathematics, 2014.

• Rice University

– BA Mathematics and Philosophy, May 2005. Cum Laude.

HONORS AND AWARDS

• Recipient of JSPS Short-term Postdoctoral Fellowship to work with Masahiro Shioyaat the University of Tsukuba, October 2014–September 2015.• Invited Scientific Researcher at the Fields Institute, Toronto, October–November 2012.• GAANN Fellowship, 2008–2010. $60,000. (Graudate Assistantships for Areas of Na-

tional Need, a competitive fellowship funded by the US Department of Education.)• National Merit Scholar, 2001.

PAPERS

• Coherent forests (submitted).• Dense ideals and cardinal arithmetic (submitted).• Ulam’s measure problem and saturation properties of ideals (in preparation).

CONFERENCE TALKS

• “Applications of the anonymous collapse.” Logic in Southern California Meeting. UCI.May 10, 2014.• “Applications of the anonymous collapse.” 15th Annual Graduate Student Conference

in Logic. University of Wisconsin, Madison. April 26, 2014.• “Ulam’s measure problem, saturated ideals, and cardinal arithmetic.” Joint Mathe-

matics Meeting. Baltimore, MD. January 16, 2014.• “Measurability properties on small cardinals.” Logic in Southern California Meeting.

UCLA. June 1, 2013.

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• “Measurability properties on small cardinals.” New York Graduate Student Conferencein Logic. CUNY Graduate Center. April 19, 2013.• “Dense ideals and small ultrapowers.” 13th Annual Graduate Student Conference in

Logic. University of Notre Dame. April 28, 2012.• “Generalization by collapse.” Association for Symbolic Logic 2012 North American

Annual Meeting. University of Wisconsin, Madison. March 31, 2012.

TEACHING EXPERIENCE

• Served as a Teaching Assistant for the following courses at UCI. Duties included leadingdiscussion section, holding office hours, writing and administering examinations, andproviding feedback to student work.

– Calculus (x6)– Math for Economists– Linear Algebra (x4)– Introduction to Abstract Math– Rings and Fields– Elementary Analysis (x2)– Complex Analysis– Modern Geometry– History of Mathematics– Introduction to Logic (x2)– Introduction to Cryptology– Probability and Stochastic Processes

• Provided private tutoring services to several clients at both collegiate and secondarylevels.

SERVICE AND LEADERSHIP EXPERIENCE

• Volunteer for “Math Fair: at the square root of fun,” hosted by the Mind ResearchInstitute, August 2014.• MathCounts volunteer, March 2014 at UC Irvine.• Associated Graduate Students Council Member, 2011-2012.• Verano Residents Council, 2012-2014. The five members of this council serve as liason

between the Verano Place graduate housing community office and the approximately1250 residents of the community. Duties include: making decisions on communityimprovements, organizing community events, advocating for residents and resolvingtheir concerns.• Gave introductory talk for undergraduates, “Definability and infinity,” to the UCI

Anteater Mathematics Club and the Chapman University MathCS Seminar. Novem-ber, 2011.

vii

ABSTRACT OF THE DISSERTATION

Measurability Properties on Small Cardinals

By

Monroe Blake Eskew

Doctor of Philosophy in Mathematics

University of California, Irvine, 2014

Professor Martin Zeman, Chair

Ulam proved that there cannot exist a probability measure on the reals for which every

set is measurable and gets either measure zero or one. He asked how large a collection of

partial 0–1 valued measures is required so that every set of reals is measurable in one of

them. Alaoglu and Erdos proved that if the continuum hypothesis holds, then countably

many measures is not enough, and Ulam asked if ℵ1 many can suffice. This question was

shown to be independent of ZFC by Prikry and Woodin. Here, we examine the analogous

questions on successor cardinals above ℵ1 and on spaces of the form Pκ(λ). We general-

ize Woodin’s consistency results to these contexts, producing models of ideals of minimal

density on various spaces starting from models of almost-huge cardinals. We show some

interactions between these ideals, cardinal arithmetic, and square principles. Then we show

that certain characterizations of a positive answer to Ulam’s question, namely the existence

of dense ideals and nonregular ideals, are equivalent on ℵ1 but not for higher cardinals. Some

tension appears in separating these properties while preserving the GCH, but we show this

is possible using structures we call “coherent forests,” about which we show several results

of independent interest. The main result is that if almost-huge cardinals are consistent, then

ZFC+GCH does not prove that the existence of dense and nonregular ideals is equivalent

for successor cardinals above ℵ1. Our methods also lead to a new result on the individual

consistency but collective inconsistency of some types of generic large cardinals.

viii

Introduction

In his 1902 thesis [27], Lebesgue considered the measure problem: Can every subset X of the

real numbers be assigned a nonnegative measure µ(X), in a way conforming to geometric

criteria, and also satisfying countable additivity? Towards a positive solution, he developed

what we now call the Lebesgue measure. A few years later in 1905, Vitali showed that

Lebesgue did not succeed in assigning a measure to every set of reals, and that a positive

solution was not achievable. In light of this, Banach and Kuratowski in 1929 [2] proposed

loosening the geometric criteria, requiring only that the measure of an interval [a, b] is |a−b|.

They proved that this was also impossible if Cantor’s Continuum Hypothesis (CH) holds.

In the following year, the Banach-Kuratowski result was strengthened by Ulam [37], who

also considered a version of the question in which every subset is given measure zero or

one. Ulam proved that, regardless of the cardinality of the continuum, a measure with such

properties could only be defined on a space whose size is “inaccessible” compared to the real

line. Ulam thus asked, for a set S of accessible cardinality such as ℵ1, ℵ2, R, 2R, etc., what

is the size of the smallest collection of countably additive two-valued partial measures, each

of which gives measure zero to single points and measure one to S, such that every subset

of S is measurable with respect to one of these measures? Strengthening his result that one

is not enough, he proved that finitely many do not suffice either. Considering measures on

ℵ1, Alaoglu and Erdos proved that countably many is still too few [10]. So Ulam asked, is

ℵ1 many measures enough [11]?

1

As it turns out, all of these questions touched upon the logical independence phenomena

discovered by Godel [15]. Many of them could not be settled by the Zermelo-Fraenkel axioms

with Choice (ZFC), but establishing this required the use of principles that substantially

transcend ZFC. Following the groundwork laid by Godel, set theory gradually established

a linear hierarchy of principles known as Large Cardinal Axioms, that empirically seem to

be able to gauge the strength of any axiomatic system. System A is said to be stronger

than system B when the consistency of B can be derived from the assumption that A is

consistent. Establishing the independence of Ulam’s questions required traveling far up this

hierarchy.

Many set theorists have viewed the large cardinal axioms as natural extensions of ZFC.

Originally, Godel and others had hoped that these axioms would be able to settle Hilbert’s

First Problem, whether CH is true [16]. After the development of the method forcing by

Cohen [7], and elaborations by Levy, Solovay and others [28], these hopes were dashed. It

was found that while these axioms have much to say about the consistency of various theo-

ries, they have relatively little direct influence on propositions about ordinary mathematical

objects like the real line. However, as advanced by Foreman [13], there is a more general class

of principles known as Generic Large Cardinals that fit under the same broad conceptual

framework as the traditional large cardinals, and these more general principles have a much

stronger influence on ordinary mathematical objects. Certain answers to Ulam’s question

about families of partial measures end up fitting into this category.

Ulam asked how many countably additive two-valued partial measures it takes to collectively

measure all subsets of ℵ1. A generalized version of this question is to take a set Z and ask

how large a family of two-valued partial measures is required to collectively measure all

subsets of Z with some additional requirements on the family, such as stipulating that they

are all κ-additive for some cardinal κ, or that they satisfy other structural properties like

normality and fineness. To any measure there is an associated ideal of measure zero sets,

2

and when the measure is κ-additive, we say the ideal is κ-complete. Sets not in a given ideal

I are called I-positive. An ideal I on a set Z is called κ-dense when there is a collection

{Aα : α < κ} of I-positive sets such that for every I-positive B ⊆ Z, there is some α < κ

such that Aα \B ∈ I; in other words Aα is contained in B except for a negligible part. It is

not hard to see that the existence of a κ-complete, κ-dense ideal on Z is equivalent to the

existence of a family of partial two-valued κ-additive measures {µα : α < κ} such that every

A ⊆ Z is either measure zero for all µα or measure one for some µα, a strengthening of Ulam’s

requirement. If an ideal is both κ-complete and κ-dense, then passing to a forcing extension

reveals properties of κ closely resembling the definitional properties of the traditional large

cardinals, hence the phrase “generic large cardinal.”

In this work, we establish the consistency of small cardinals possessing various kinds of

generic largeness properties and explore the interrelationships between these properties and

some more standard propositions of infinitary combinatorics. Our consistency results start

from traditional large cardinal assumptions that lie between almost-huge and huge.

Chapter 1 lays out the necessary preliminaries about ideals, forcing, and elementary embed-

dings. Many proofs are deferred to well-known textbooks.

Chapter 2 shows how to obtain models of normal and fine, κ-complete λ-dense ideals ideals

on Pκ(λ) where κ is a successor cardinal, giving positive answers to many generalizations

of Ulam’s problem. For κ = ℵ1, these ideals have in some sense the maximal saturation

property, but for higher successor cardinals, there are more structural possibilities. A key to

this construction is a certain “universal” boolean algebra we dub “the anonymous collapse.”

Its flexibility enables saturated ideals on Pκ(λ) for a fixed successor κ and many values of

λ simultaneously. It also has several interesting applications without the use of hypotheses

near the strength of almost-huge cardinals. For example, we can use it to produce many

models with the same cardinals and same reals, but very different higher-order combinatorial

properties of the continuum, a phenomenon belied by the phrase “cardinal invariants of the

3

continuum.”

Chapter 3 explores some consequences of the existence of these generic large cardinals for

cardinal arithmetic, square principles, and an old conjecture from model theory about the

size of ultrapowers. We answer two open questions posed by Foreman in [13]. These explo-

rations lead to an interesting limitation regarding successors of singular cardinals, showing

the optimality of some aspects of the consistency results.

Chapter 4 focuses on several properties related to a positive solution to Ulam’s question,

which Taylor [33] proved equivalent relative to ℵ1. We explore the extent to which the

arguments for Taylor’s theorem generalize to higher cardinals. The key notion is that of

“regularity” of an ideal, and we show that under the Generalized Continuum Hypothesis

(GCH), most degrees of regularity are equivalent. We also show that under GCH, a positive

solution to the generalized Ulam problem via normal ideals is equivalent to the existence of a

dense ideal, implying that the generalized Ulam problem has a negative answer at successors

of singular cardinals under GCH.

Chapter 5 gives consistency results relative to generic large cardinals that were proved con-

sistent relative to almost-huge cardinals in Chapter 2. The modular nature of this chapter

means that it can piggyback on possible future results that may reduce upper bounds on the

consistency strength of dense ideals above ℵ1. We start with a generalization of Foreman’s

Duality Theorem, fixing a minor error in [14]. Using this, we separate the existence of dense

ideals from nonregular ideals above ℵ1, showing that Taylor’s theorem is indeed specific to

ℵ1. Although the equivalence of a positive answer to the generalized Ulam problem and

dense ideals is open, we show they can never be separated via this technique. We also use

the Duality Theorem to show that strong forms of generic supercompactness are compatible

with square holding globally, in contrast to traditional supercompactness. Finally, we apply

these techniques to show that some types of generic large cardinals cannot coexist in one

model of set theory, strengthening a result of Woodin.

4

There is an apparent tension between the technique used to separate density and nonregu-

larity, and the preservation of GCH. Chapter 6 is aimed at resolving this problem, showing

ultimately that if almost-huge cardinals are consistent, then ZFC+GCH does not prove a

generalization of Taylor’s theorem to cardinals above ℵ1. We arrive at this through an in-

vestigation of structures dubbed “coherent forests,” given their connection to the trees of

infinitary combinatorics. The notions of being Aronszajn and Suslin carry over from trees

to forests, and we explore several ways of obtaining large Aronszajn and Suslin forests. We

show that large coherent Aronszajn forests can be constructed within ZFC and use the P-

ideal dichotomy to show the optimality of some of these results. Then we give three ways

of forcing large coherent Suslin forests. The first is a modification of Jech’s method of forc-

ing by local approximations [19], and the second generalizes the well-known argument of

Todorcevic that a Cohen real adds a Suslin tree [35]. The third method uses a guessing

principle, which we show consistent from a Mahlo cardinal, that plays a similar role to dia-

mond in the construction of Suslin trees. This allows a large Suslin forest to be created by

a relatively small forcing. This feature leads to models with the right kind of dense ideals

and large Suslin algebras existing simultaneously, allowing the techniques of Chapter 5 to

be applied to achieve the main result.

5

Chapter 1

Preliminaries

We start by reviewing some essential facts about ideals, forcing, and elementary embeddings.

Many of these results are folklore, and when proofs are omitted, they may be found in [13],

[21], [22], or [25].

1.1 Basic combinatorics of ideals

Let Z be any set. An ideal I on Z is a collection of subsets of Z closed under taking subsets

and pairwise unions. If κ is a cardinal, I is called κ-complete if it is also closed under unions

of size less than κ. “Countably complete” is taken as synonymous with “ω1-complete.” I is

called nonprincipal if {z} ∈ I for all z ∈ Z, and proper if Z /∈ I. Hereafter we will assume

all our ideals are nonprincipal and proper.

Let X =⋃Z. I is called fine if for all x ∈ X, {z : x /∈ z} ∈ I. I is called normal if for

any sequence 〈Ax : x ∈ X〉 ⊆ I, the “diagonal union” {z : ∃x(x ∈ z ∈ Ax)} is in I. It is

well-known that I is normal iff for any A ∈ P(Z) \ I and any function f on A such that

f(z) ∈ z for all z ∈ A, there is an x such that f−1(x) /∈ I.

6

To fix notation, let I∗ = {Z \ A : A ∈ I} (the I-measure one sets), I+ = P(Z) \ I (the

I-positive sets), x = {z : x ∈ z}, and denote diagnoal unions by ∇x∈XAx. Note that

∇x∈XAx =⋃x∈X x ∩ Ax.

Proposition 1.1. Let I be a normal and fine ideal on Z ⊆ P(X). Let κ be a cardinal

and let {xα : α < κ} be distinct elements of X. Then I is κ-complete iff for all β < κ,⋂α<β xα ∈ I∗.

Proof. If I is κ-complete, then by fineness xα ∈ I∗ for any α, so⋂α<β xα ∈ I∗ for any β < κ.

For the other direction, suppose that {Aα : α < β < κ} ⊆ I, but A =⋃α<β Aα ∈ I+. Then

by hypothesis, B = A ∩ (⋂α<β xα) ∈ I+. Let f : B → X be defined by f(z) = xα, where α

is the least ordinal such that z ∈ Aα. By normality, some Aα ∈ I+, a contradiction.

The following basic fact seems to have been previously overlooked–see, for example, the

hypotheses of several theorems in [13] and [14].

Proposition 1.2. All normal and fine ideals are countably complete.

Proof. Let I be a normal and fine ideal on Z ⊆ P(X). By the above, it suffices to find an

infinite set {xn : n < ω} ⊆ X such that⋂xn ∈ I∗. Since I is proper and nonprincipal, X is

infinite. We show that any infinite set of distinct elements of X suffices.

Let {xn : n < ω} be distinct elements of X. Suppose the contrary, that B = {z : {xn : n <

ω} * z} ∈ I+. By fineness, B ∩ x0 ∈ I+. For each z ∈ B ∩ x0, let nz be the largest integer

such that {x0, ..., xnz} ⊆ z. Let f : B ∩ x0 → X be defined by f(z) = xnz . By normality,

there is an n such that C = f−1(xn) ∈ I+. Then for all z ∈ C, xn+1 /∈ z. This contradicts

fineness.

Proposition 1.3. If I is a normal, fine, κ-complete ideal on Z ⊆ Pκ(κ), then κ ∈ I∗.

7

Proof. Suppose A = {z ∈ Z : z is not an ordinal} ∈ I+. Let f : A→ κ be such that f(z) is

the least α ∈ z such that α * z. Then for some α, f−1(α) ∈ I+. However, {z : α ⊆ z} ∈ I∗

by fineness and κ-completeness.

Lemma 1.4. If κ = µ+ and I is a κ-complete, fine ideal on Z = Pκ(X), then every A ∈ I+

can be split into |X| many disjoint I-positive sets.

Proof. We use a generalization of Ulam matrices. For each z ∈ Z, let fz : z → µ be an

injection. For α < µ, x ∈ X, let Mαx = {z ∈ x : fz(x) = α}. For x 6= y, Mα

x ∩Mαy = ∅. For

each x ∈ X, x =⋃α<µM

αx . If A ∈ I+, then by κ-completeness, ∀x∃α(A ∩Mα

x ∈ I+). If |X|

is regular, then there is some α < µ such that A ∩Mαx ∈ I+ for |X| many x ∈ X.

Otherwise, for each β < |X|, there is some α < µ such that |{x ∈ X : A ∩Mαx ∈ I+}| ≥ β,

since X =⋃α<µ{x : A ∩Mα

x ∈ I+}. Pick some sequence 〈βi : i < cf(|X|)〉 converging to

|X|, pick some α < µ such that |{x ∈ X : A∩Mαx ∈ I+}| ≥ cf(|X|), and enumerate the first

cf(|X|) elements as 〈xi : i < cf(|X|)〉. For each xi, apply the above argument to pick some

splitting of A∩Mαxi

into βi many disjoint I-positive sets. This gives a splitting of A into |X|

many disjoint I-positive sets.

Alaoglu and Erdos set the stage for Ulam’s question by providing an important limitation.

Taylor [34] generalized their result to show that for any collection {Iα : α < µ} of µ+-

complete ideals on µ+, there is an A ⊆ µ+ that is nonmeasurable for each Iα. A similar

result holds for normal and fine ideals on Pκ(λ):

Theorem 1.5. Suppose {Iα : α < η < λ} is a collection of normal and fine ideals on

Z ⊆ P(λ) such for each α < η and each A ∈ I+α , A can be split into η+ many disjoint Iα-

positive sets. Then there is a sequence of disjoint sets {Aα : α < η} such that each Aα ∈ I+α

and a set B ⊆ Z that is nonmeasurable for all Iα.

Proof. For each α < η, let {Aαβ : β < η+} be a collection of disjoint Iα-positive sets. Define

8

f(α) to be the least γ ≤ α such that Aγβ ∈ I+α for η+ many β. Let δ < η+ be such

that (∀α < η)(∀γ < f(α))(∀β ≥ δ)Aγβ ∈ Iα. Recursively construct a one-to-one sequence

〈Bα : α < η〉 such that Bα = Af(α)β ∈ I+

α for some β ≥ δ. Then Bβ ∈ Iα when f(β) < f(α),

and Bβ ∩Bα = ∅ when f(β) = f(α). For each α < η, let Cα = Bα ∩ α \⋃f(β)<f(α)(Bβ ∩ β).

Cα ∈ I+α by normality, and Cα ∩ Cβ = ∅ when α 6= β. Now split each Cα into two disjoint

Iα-positive sets, C0α and C1

α. For i < 2, let Di =⋃α<η C

iα. D0 and D1 are disjoint and

Iα-positive for all α, hence nonmeasurable for all Iα.

Therefore, if κ = µ+ and λ ≥ κ, the best we could hope for is to measure every subset of

Pκ(λ) with λ many κ-complete, normal and fine ideals. Using large cardinals, we will show

that this is possible in many cases.

1.2 Forcing

A partial order P is said to be separative when p � q ⇒ (∃r ≤ p)r ⊥ q. Every partial order

P has a canonically associated equivalence relation s and a separative quotient Ps, which is

isomorphic to P if P is already separative. In most cases we will assume our partial orders

are separative. For every separative partial order P, there is a canonical complete boolean

algebra B(P) with a dense set isomorphic to P.

A map e : P→ Q is an embedding when it preserves order and incompatibility. An embedding

is said to be regular when it preserves the maximality of antichains. A order-preserving map

π : Q→ P is called a projection when π(1Q) = 1P, and p ≤ π(q)⇒ (∃q′ ≤ q)π(q′) ≤ p.

Lemma 1.6. Suppose P and Q are partial orders.

(1) G is a generic filter for P iff {[p]s : p ∈ G} is a generic filter for Ps.

9

(2) e : P→ Q is a regular embedding iff for all q ∈ Q, there is p ∈ Q such that for all r ≤ p,

e(r) is compatible with q.

(3) The following are equivalent:

(a) There is a regular embedding e : Ps → B(Qs).

(b) There is a projection π : Qs → B(Ps).

(c) There is a Q-name g for a P-generic filter such that for all p ∈ P, there is q ∈ Q

such that q p ∈ g.

(4) Suppose e : P → Q is a regular embedding. If G is a filter on P let Q/G = {q : ¬∃p ∈

G(e(p) ⊥ q)}. The following are equivalent:

(a) H is Q-generic over V .

(b) G = e−1[H] is P-generic over V , and H is Q/G-generic over V [G].

Lemma 1.7. Suppose P and Q are partial orders. B(Ps) ∼= B(Qs) iff the following holds.

Letting G, H be the canonical names for the generic filters for P,Q respectively, there is a

P-name for a function f0 and a Q-name for a function f1 such that:

(1) P f0(G) is a Q-generic filter,

(2) Q f1(H) is a P-generic filter,

(3) P G = ff0(G)1 (f0(G)), and Q H = f

f1(H)0 (f1(H)).

Proof. If ι : B(Ps) ∼= B(Qs) is an isomorphism, then we can let f0 be a P-name for ι[G], and

f1 be a Q-name for ι−1[H].

Suppose P and Q are complete boolean algebras and f0, f1 are as hypothesized. Let ι : P→ Q

be given by p 7→ ||p ∈ f1(H)||. First note that ι clearly preserves order and incompatibility.

10

The kernel of ι is trivial, since for any nonzero p ∈ P, if we take G generic with p ∈ G, then

H = f0(G) is Q-generic, and p ∈ G = f1(H).

It suffices to show that the range of ι is dense. Let q ∈ Q be arbitrary. First we claim

there is p ∈ P that forces q ∈ f0(G). If H is generic with q ∈ H, then let G = f1(H) and

let p ∈ G force q ∈ f0(G) = H. Now we claim for such p, ι(p) ≤ q. For whenever H is

generic with ι(p) ∈ H, p ∈ f1(H) = G by the definition of ι, and so by the property of p,

q ∈ f0(G) = H.

For a broader notion of “forcing equivalence,” the best that can be said in general is the

following:

Lemma 1.8. Suppose P and Q are partial orders.

(1) If e : P → Q is a regular embedding, and any Q-generic H yields V [H] = V [e−1[H]],

then there is a predense set A ⊆ B(Qs) such that B(Ps) ∼= B(Qs) � a for all a ∈ A.

(2) P and Q yield the same generic extensions iff for a dense set of p ∈ P, there is q ∈ Q

such that B(Ps) � p ∼= B(Qs) � q and vice versa.

Proof. For both claims, we will assume P and Q are complete boolean algebras.

For (1), if V [H] = V [e−1[H]] for any Q-generic H, then P must force that the quotient

Q/e[G] is atomic. Hence there is an isomorphism ι : P ∗ A → Q extending e, where A is a

P-name for an atomic boolean algebra. The set of elements (1, a) for a a name for an atom

is predense, and Q � ι(1, a) is isomorphic to P for any name for an atom a.

For (2), suppose P and Q yield the same extensions, and let p0 ∈ P be arbitrary. Let G ⊆ P

be generic with p0 ∈ G. Since every P-generic extension is a Q-generic extension, there is a

P-name h for a Q-generic filter such that 1 G ∈ V [h]. There must be some q0 ∈ Q such

11

that ||q ∈ h|| ∧ p0 6= 0 for all q ≤ q0; otherwise the set of q which p0 forces cannot be in

h is dense, contradicting the genericity of h. The map e : q 7→ ||q ∈ h|| ∧ p0 is a regular

embedding of Q � q0 into P � ||q0 ∈ h|| ∧ p0, since for any maximal antichain A ⊆ Q � q0 and

any generic G ⊆ P with ||q0 ∈ h|| ∧ p0 ∈ G, there is some q ∈ A such that q ∈ hG. Thus the

hypotheses of (1) are satisfied, and Q � q0∼= P � p1 for some p1 ≤ p0. Switching the roles of

P and Q gives the “vice versa” conclusion. The converse is trivial.

A partial order P is said to be κ-distributive if for any collection of maximal antichains in

P, {Aα : α < β < κ}, there is a maximal antichain A such that A refines Aα for all α < β.

P is called (κ, λ)-distributive if the same holds restricted to antichains of size ≤ λ. Forcing

with P adds adds no new functions from any α < κ to λ iff B(P) is (κ, λ)-distributive.

A strictly stronger property than distributivity is strategic closure. For a partial order P

and an ordinal α, we define a game Gα(P) with two players Even and Odd. Even starts by

playing some element p0 ∈ P. At successor stages β + 1, the next player must play some

element pβ+1 ≤ pβ. Even plays at limit stages β if possible, by playing a pβ that is ≤ pγ

for all γ < β. If Even cannot play at some stage below α, the game is over and Odd wins;

otherwise Even wins. We say that P is α-strategically closed if for every p ∈ P, Even has

a winning strategy with first move p. Note that under this definition, every partial order is

trivially ω-strategically closed.

A stronger property that κ-strategic closure is κ-closure. P is κ-closed when any descending

chain of length less than κ has a lower bound. P is κ-directed closed when any directed set

of size < κ has a lower bound.

For any partial order P, the saturation of P, sat(P), is the least cardinal κ such that every

antichain in P has size less than κ. Erdos and Tarski [12] proved that sat(P) is always regular.

The density of P, d(P), is the least cardinality of a dense subset of P. Clearly sat(P) ≤ d(P)+

for any P. We say P is κ-saturated if sat(P) ≤ κ, and P is κ-dense if d(P) ≤ κ. A synonym

12

for κ-saturation is the κ chain condition (κ-c.c.).

The properties of distributivity, strategic closure, saturation, and density are robust in the

sense that they are absolute between P and B(P) for any separative partial order P, and

often inherited by intermediate forcings:

Lemma 1.9. Suppose e : P→ Q is a regular embedding and κ is a cardinal.

(1) If Q is κ-strategically closed, then so is P.

(2) Q is κ-distributive iff P is κ-distributive and P Q/G is κ-distributive.

(3) Q is κ-saturated iff P is κ-saturated and P Q/G is κ-saturated.

(4) Q is κ-dense iff P is κ-dense and P Q/G is κ-dense.

Proof. We prove only (4). Suppose first that Q is κ-dense, and let {qα : α < κ} witness.

Since B(Q) ∼= B(P ∗ Q/G, we can pick for each qα some (pα, rα) ≤ qα. Then {pα : α < κ}

is dense in P. If G ⊆ P is generic, let r ∈ Q/G be arbitrary. For any p ∈ P, there is

(pα, rα) ≤ (p, r), so for some pα ∈ G, pα rα ≤ r. Now suppose {pα : α < κ} is dense in P,

and {rβ : β < κ} is dense in Q/G. For any q ∈ Q, let (p, r) ≤ q. By density, there is some

pα < p and some β < κ such that pα rβ ≤ r.

For any forcing P and any P-name X for a set of ordinals, there is a canonically associated

complete subalgebra AX ⊆ B(P) that captures X. It is the smallest complete subalgebra

containing all elements of the form ||α ∈ X|| for α an ordinal. AX has the property that

whenever G ⊆ P is generic, XG and G ∩ AX are definable from each other using the pa-

rameters B(P) and its powerset, as computed in the ground model. In this case, we have

V [XG] = V [G ∩ AX ].

13

1.3 Forcing with ideals

Proofs of the following facts can be found in [13]. If I is an ideal on Z, say A ∼I B if the

symmetric difference A∆B is in I. Let [A]I denote the equivalence class of A mod ∼I . The

equivalence classes form a boolean algebra under the obvious operations, which we denote

by P(Z)/I. Normality ensures a certain amount of completeness of the algebra:

Proposition 1.10. Suppose I is a normal and fine ideal on Z ⊆ P(X). If {Ax : x ∈ X} ⊆

P(Z), then ∇Ax is the least upper bound of {[Ax]I : x ∈ X} in P(Z)/I.

If we force with this algebra, we get a generic ultrafilter G on Z extending I∗. We can form

the ultrapower V Z/G. If this ultrapower is well-founded for every generic G, then I is called

precipitous. A combinatorial characterization of precipitousness is given by the following:

Theorem 1.11 (Jech-Prikry). I is a precipitous ideal on Z iff the following holds: For any

sequence 〈An : n < ω〉 ⊆ P(I+), such that for each n,

(1) Bn = {[a]I : a ∈ An} is a maximal antichain in P(Z)/I,

(2) Bn+1 refines Bn,

there is a function f with domain ω such that for all n, f(n) ∈ An, and⋂n<ω f(n) 6= ∅.

For an ideal I, the saturation, density, distributivity, and strategic closure of I refers to that

of the corresponding boolean algebra. The next proposition is immediate from Theorem 1.11:

Proposition 1.12. If I is an ω1-complete, ω1-distributive ideal, then I is precipitous.

Proposition 1.13. Suppose I is a κ-complete precipitous ideal on Z, and I is nowhere

κ+-complete. Let G be P(Z)/I-generic, and let j : V → M be the associated elementary

embedding, where M is the transitive collapse of V Z/G. Then the critical point of j is κ.

14

Proposition 1.14. Let I be an ideal Z ⊆ P(X). Then I is normal and fine iff 1 P(Z)/I

[id] = j[X].

Proposition 1.15. Suppose I is an ideal on Z ⊆ P(X). If I is κ-complete and κ+-saturated,

or if I is normal, fine, and |X|+-saturated, then every antichain in P(Z)/I has a system of

pairwise disjoint representatives.

Proof. If I is κ-complete, and {Aα : α < κ} is an antichain, replace each Aα with Aα \⋃β<αAβ. If I is normal and fine, and {Ax : x ∈ X} is an antichain, replace Ax by Ax ∩ x \⋃y 6=xAy ∩ y.

Theorem 1.16. Suppose I is a countably complete ideal on Z, and every antichain in

P(Z)/I has a system of pairwise disjoint representatives. Then:

(1) I is precipitous.

(2) P(Z)/I is a complete boolean algebra.

(3) If G is generic over P(Z)/I, j : V → M is the associated embedding, and j[λ] ∈ M ,

then M is closed under λ-sequences from V [G].

1.4 Elementary embeddings

Lemma 1.17. Suppose M and N are models of ZF−, j : M → N is an elementary embed-

ding, P ∈M is a partial order, G is P-generic over M , and H is j(P)-generic over N . Then

j has a unique extension j : M [G]→ N [H] with j(G) = H iff j[G] ⊆ H.

Proof. If j[G] ⊆ H, the only possible choice is to let j(τG) = j(τ)H for all P-names τ .

If M [G] |= ϕ(τG1 , ..., τGn ), then for some p ∈ G, p ϕ(τ1, ..., τn). We have j(p) ∈ H and

15

j(p) ϕ(j(τ1), ..., j(τn)), so N [H] |= ϕ(j(τ1)H , ..., j(τn)H). Conversely, if j : M [G] → N [H]

is elementary, extends j, and has j(G) = H, then for all p ∈ G, j(p) ∈ H by elementarity.

Lemma 1.18. Suppose M , N are transitive models of ZFC with the same ordinals, and

j : M → N is an elementary embedding. Then either j has a critical point, or j is the

identity and M = N .

Proof. Suppose j is the identity map on ordinals. Let x be a set of minimal rank in M

such that j(x) 6= x. There is some ordinal κ ∈ M such that x = {xα : α < κ}. Then

j(x) = {j(xα) : α < κ} = x by the minimality of x, so j is the identity. To show M = N ,

note that for all ordinals α ∈M , j(V Mα ) = V N

α = V Mα .

16

Chapter 2

Dense ideals from large cardinals

Here we show that it is consistent relative to an almost-huge cardinal that there is a normal,

κ-complete, λ-dense ideal on Pκ(λ), where κ is the successor of a regular cardinal µ, and

λ ≥ κ is regular, for many particular choices for µ, λ. We also show that relative to a

super-almost-huge cardinal, there can exist a successor cardinal κ such that for every regular

λ ≥ κ, there is a normal, κ-complete, λ-dense ideal on Pκ(λ). This generalizes a theorem

of Woodin about the relative consistency of an ℵ1-dense ideal on ℵ1, and has the following

additional advantages: (1) An explicit forcing extension is taken, rather than an inner model

of an extension. (2) Careful constructions within a model where the axiom of choice fails,

as presented in [13], are avoided.

Let us first recall the essential facts about almost-huge cardinals (see [22], Theorem 24.11).

A cardinal κ is almost-huge if there is an elementary embedding j : V → M with critical

point κ, such that M<j(κ) ⊆M .

Theorem 2.1. The following are equivalent:

(1) κ carries an almost-huge embedding j such that j(κ) = δ.

17

(2) δ is inaccessible, and there is a sequence 〈Uα : κ ≤ α < δ〉 such that:

(a) each Uα is a normal, κ-complete ultrafilter on Pκ(α),

(b) for α < β, Uα = {A ⊆ Pκ(α) : {z ∈ Pκ(β) : z ∩ α ∈ A} ∈ Uβ}, and

(c) for all α < δ and all f : Pκ(α) → κ such that {z : f(z) ≥ ot(z)} ∈ Uα, there is β

such that α ≤ β < δ and {z : f(z ∩ α) = ot(z)} ∈ Uβ.

Furthermore, if a system as in (2) is given, the direct limit model and embedding witness the

almost-hugeness of κ with target δ.

A system as in (2) will be called an almost-huge tower. Almost-huge towers capture almost-

hugeness in a minimal way:

Corollary 2.2. If κ has an almost-huge tower of height δ, and j : V →M is the embedding

derived from the tower, then we have δ < j(δ) < δ+, and j[δ] is cofinal in j(δ).

Proof. For each α < δ, let Mα be the transitive collapse of V Pκ(α)/Uα, and let jα : V →Mα

and kα : Mα → M be the canonical embeddings, with j = kα ◦ jα. Since δ is inaccessible,

jα(κ) < δ and jα(δ) = δ for each α < δ.

If γ < j(δ), then there are some α, β < δ such that kα(β) = γ. Thus there are only δ ordinals

below j(δ). Also, there is η < δ such that jα(η) > β, so j(η) > γ, and thus j[δ] is cofinal in

j(δ).

A super-almost-huge cardinal is a cardinal κ such that for all λ ≥ κ, there is an almost huge

tower of height ≥ λ. The next result follows from considering the set of closure points under

witnesses to property (c) in the tower characterization.

Corollary 2.3. If κ has an almost-huge tower of height δ, and δ is Mahlo, then Vδ |=

ZFC + κ is super-almost-huge, and for stationary many α < δ, Vα |= ZFC + κ is super-

almost-huge.

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There is a vast gap in strength between almost-huge and huge:

Theorem 2.4. If κ is a huge cardinal, then there is a stationary set S ⊆ κ such that for all

α < β in S, α has an almost-huge tower of height β.

Proof. Suppose j : V → M is an elementary embedding with critical point κ, j(κ) = δ,

and M δ ⊆ M . Then κ carries an almost-huge tower ~U of length δ, and ~U ∈ M . Let F be

the ultrafilter on κ defined by F = {X ⊆ κ : κ ∈ j(X)}. Let A = {α < κ : α carries an

almost-huge tower of height κ}. Since κ ∈ j(A), A ∈ F . Now let c : κ2 → 2 be defined by

c(α, β) = 1 if α carries and almost-huge tower of height β, and c(α, β) = 0 otherwise. By

Rowbottom’s theorem, let H ∈ F be homogeneous for c. We claim c takes constant value 1

on H. For if α ∈ A ∩H, then {α, κ} ∈ [j(A ∩H)]2, and j(c)(α, κ) = 1.

2.1 Layering and absorption

Definition. We will call a partial order P (µ, κ)-nicely layered when there is a collection L

of regular suborders of P such that:

(1) for all Q ∈ L, Q is µ-closed and has size < κ,

(2) for all Q0,Q1 ∈ L, if Q0 ⊆ Q1, then Q0 Q1/G is µ-closed, and

(3) for all P-names f for a function from µ to the ordinals, and all Q0 ∈ L, there is an

Q1 ∈ L and an Q1-name g such that Q0 ⊆ Q1, and P f = g.

We will say P is (µ, κ)-nicely layered with collapses, (µ, κ)-NLC, when additionally for all

α < κ and all Q0 ∈ L, there is Q1 ∈ L such that Q0 ⊆ Q1, Q0 |Q1/G| ≥ |α|, and

Q1 |Q1| = µ.

19

Proposition 2.5. If L witnesses that P is (µ, κ)-nicely layered, then P is κ-c.c. and⋃L is

dense in P.

Proof. Suppose that {pα : α < κ} ⊆ P is a maximal antichain. Let f be a name of a function

with domain {0} such that f(0) = α iff pα ∈ G. There cannot be a regular suborder Q of

size < κ and a Q-name g that is forced to be equal to f , since such a g would have < κ

possible values for its range.

Similarly, let p ∈ P be arbitrary, and let {pα : α < δ} be a maximal antichain with p = p0.

Let f be a name of a function with domain {0} such that f(0) = α iff pα ∈ G. If Q is a

regular suborder and g is a Q-name such that P f = g, then there is some q ∈ Q forcing

g(0) = 0, so q ≤ p.

Lemma 2.6 (Folklore). If P is a µ-closed partial order such that P |P| = µ, then B(P) ∼=

B(Col(µ, |P|)).

Proof. Pick a P-name f for a bijection from µ to G. We build a tree T ⊆ P that is isomorphic

to a dense subset of Col(µ, |P|), and show that it is dense in P. Each level will be a maximal

antichain in P. Let the first level T0 = {1P}. If levels {Tβ : β < α+1} are defined, below each

p ∈ Tα, pick a |P|-sized maximal antichain of conditions deciding f(α), and let Tα+1 be the

union of these antichains. If {Tβ : β < λ} is defined up to a limit λ, pick for each descending

chain b through the previous levels, a |P|-sized maximal antichain of lower bounds to b, and

set Tλ equal to the union of these anithchains. It is easy to check that Tλ is a maximal

antichain. Let T =⋃α<µ Tα. To show T is dense, let p ∈ P. Let q ≤ p be such that for some

α < µ, q f(α) = p. q is compatible with some r ∈ Tα+1. Since r decides f(α) and forces

it in G, r ≤ p.

Lemma 2.7. Suppose µ < κ are regular, and P is (µ, κ)-NLC. If G is P-generic over V ,

then there is a forcing R ∈ V [G] such that R adds a filter H ⊆ Col(µ,< κ) which is generic

over V and such that (Ordµ)V [G] = (Ordµ)V [H].

20

Proof. Let L witness the (µ, κ)-NLC property. First note that this implies α<µ < κ for all

α < κ. In V [G], let R be the collection of filters h ⊆ Col(µ,< α) for α < κ which are generic

over V , such that for some Q ∈ L, V [h] = V [G ∩Q]. The ordering is end-extension.

Let h ∈ R with Q0 ∈ L a witness, and let and α < κ be arbitrary. Let α < β < κ and

Q1 ⊇ Q0 in L be such that in V [h], |Q1/(G ∩ Q0)| = |β|, and Q1 collapses β to µ. By the

definition and Lemma 2.6, Q1/(G∩Q0) is equivalent in V [h] to Col(µ, β), which is equivalent

to the < µ-support product of Col(µ, γ) for α ≤ γ ≤ β. The filter G ∩Q1 therefore gives a

filter h′ ⊇ h on Col(µ,< β + 1) that is generic over V , with V [h′] = V [Q1 ∩G].

Let h ∈ R with Q0 ∈ L a witness, and let f : µ → Ord in V [G] be arbitrary. By the

definition of (µ, κ)-NLC, we can find some Q1 ⊇ Q0 in L such that f ∈ V [G ∩ Q1]. By the

previous paragraph, we may find Q2 ⊇ Q1 in L equivalent to some Col(µ,< α), and some

filter h′ ⊆ Col(µ,< α) generic over V , extending h, and such that V [G ∩Q2] = V [h′].

So if F is generic over R, let H =⋃h∈F h. By the above arguments, the rank of H is κ.

Since Col(µ,< κ) is κ-c.c., H is generic, since any maximal antichain from V intersects some

h ∈ F . Also by the above arguments, any f : µ→ Ord in V [G] is in V [H]. Conversely, any

f : µ→ Ord in V [H] lives in some V [h] with h ∈ R, so is in V [G].

2.1.1 The anonymous collapse

Let κ be a regular cardinal whose regularity is preserved by a forcing P. Let A(P) be the

complete subalgebra of B(P∗Add(κ)) generated by the canonical name for the Add(κ)-generic

set. More precisely, if e : P ∗ Add(κ) → B(P ∗ Add(κ)) is the canonical dense embedding,

A(P) is completely generated by the elements of the form e(〈1, ˙{〈α, 1〉}〉).

In the case that α<µ < κ for all α < κ and P = Col(µ,< κ), denote A(P) by A(µ, κ), and

write B(µ, κ) for B(Col(µ,< κ) ∗ Add(κ)).

21

Lemma 2.8. If P is (µ, κ)-NLC, and H ⊆ A(P) is generic over V , then B(P ∗ Add(κ))/H

is κ-distributive in V [H].

Proof. V [H] = V [XH ] for some canonically associated X ⊆ κ, and by forcing with B(P ∗

Add(κ))/H over V [H], we recover a filter G ∗XH for P ∗ Add(κ), generic over V .

If G ∗ X is P ∗ Add(κ)-generic over V , then X codes all subsets of µ that live in V [G].

By the definition of (µ, κ)-NLC, every z ∈ (Ordµ)V [G] occurs in some submodel of the form

V [G ∩ Q], where Q is isomorphic to Col(µ, α) for some α < κ. Thus z ∈ V [y] for some

y ⊆ µ in V [G], so (Ordµ)V [X] ⊇ (Ordµ)V [G]. Since Add(κ) adds no µ-sized sets of ordinals,

(Ordµ)V [G] = (Ordµ)V [G∗X] ⊇ (Ordµ)V [X]. Thus B(P ∗ Add(κ))/H is κ-distributive.

Lemma 2.9. Let V be a countable transitive model of ZFC (or just assume generic extensions

are always available), and assume VP κ is regular. If X ⊆ κ, the following are equivalent:

(1) X is A(P)-generic over V .

(2) There is G ⊆ P such that G is generic over V , and X is Add(κ)-generic over V (P0),

where P0 = Pκ(κ)V [G].

Proof. If X is A(P)-generic then force with B(P ∗Add(κ))/HX over V [X], obtaining G such

that G ∗ X is P ∗ Add(κ)-generic over V . Then X is Add(κ)-generic over V [G], and since

Add(κ)V [G] = Add(κ)V (P0), X is Add(κ)-generic over V (P0).

Suppose G ⊆ P is generic over V , and X is Add(κ)-generic over V (P0), but not A(P)-generic

over V . Then some p ∈ Add(κ)V (P0) forces this with dom(p) = α < κ, and X � α = p. Take

Y ⊆ κ such that Y � α = p that is Add(κ)-generic over the larger model V [G]. Then Y is

A(P)-generic over V , and V (P0)[Y ] can see this, but this contradicts the property of p. So

X was A(P)-generic over V .

22

Theorem 2.10. For any P that is is (µ, κ)-NLC, there is an isomorphism ι : A(P)→ A(µ, κ)

such that ι(||α ∈ X||A(P)) = ||α ∈ X||A(µ,κ) for all α < κ.

Proof. Let X be A(P)-generic over V . There is a κ-distributive forcing over V [X] to get G

such that G ∗X is P ∗ Add(κ)-generic over V . By Lemma 2.7, we can do further forcing to

obtain H ⊆ Col(µ,< κ) generic over V such that (Ordµ)V [H] = (Ordµ)V [G]. By Lemma 2.9,

X is also A(µ, κ)-generic over V .

Conversely, every A(µ, κ)-generic X is A(P)-generic. For suppose X is a counterexample.

Then there is some (p, q) ∈ Col(µ,< κ)∗Add(κ) such that (p, q) X is not A(P)-generic over

V . Let Y be any A(P)-generic set, and let P0 = P(µ)V [Y ]. By the above, Y is A(µ, κ)-generic

over V . Thus we can force over V [Y ] to get H ⊆ Col(µ,< κ) such that H ∗Y is generic over

V . By the homogeneity of the Levy collapse, there is some automorphism π ∈ V such that

p ∈ π[H] = H ′. By the homogeneity of Cohen forcing, there is some automorphism σ in

V (P0) such that σ[Y ] is a generic Y ′ such that Y ′ � dom(qH′) = qH

′. Y ′ is also A(P)-generic

over V . However, (p, q) ∈ H ′ ∗ Y ′, so we have a contradiction.

This implies that we have a canonical correspondence between A(P)- and A(µ, κ)-generic

filters, i.e. definable functions f, g such that for any generic H for A(P), f(H) is the generic

for A(µ, κ) computed from XH , and vice versa, and g(f(H)) = H. For p ∈ A(P), put

ι(p) = ||p ∈ g(H)||A(µ,κ). It is easy to see that ι is a complete embedding. For any q ∈ A(µ, κ),

there is p ∈ A(P) forces that q ∈ f(H). Thus if H is generic for A(µ, κ) and ι(p) ∈ H, then

p ∈ g(H), so q ∈ f(g(H)) = H, hence ι(p) ≤ q. The range of ι is dense, so it is an

isomorphism. By the way we construct f and g, ι(||α ∈ X||A(P)) = ||α ∈ X||A(µ,κ).

This machinery has some interesting applications to the absoluteness of some properties of

a given powerset. First, it is easy to see for regular µ < κ such that α<µ < κ for all α < κ,

Col(µ,< κ)×Add(µ, λ) is (µ, κ)-NLC for every λ. Thus ifX isA(µ, κ)-generic, then for any λ,

we may further force to obtain a model which is a (Col(µ,< κ)×Add(µ, λ))∗Add(κ)-generic

23

extension with the same Ordµ. Taking inner models given by such Col(µ,< κ)×Add(µ, λ)-

generic sets, we produce many models with the same cardinals and same P(µ), each assigning

a different cardinal value for 2µ. For example, if we add ω1 Cohen reals to any model of M

of ZFC, this is the same as forcing with Col(ω,< ω1). There is for each uncountable ordinal

α ∈ M , a generic extension with the same reals and same cardinals, in which it appears we

have added α many Cohen reals.

By using weakly compact cardinals, we can get even more dramatic examples. If κ is weakly

compact, every κ-c.c. partial order captures small sets in small factors. To show this, first

consider a partial order P of size κ. We can code P as A ⊆ κ, and by weak compactness,

there is some transitive elementary extension (Vκ,∈, A) ≺ (M,∈, B). If µ < κ, then any

P-name for function f : µ→ Ord has an equivalent name τ ∈ Vκ by the κ-c.c. Since A ∈M

and M sees A as a regular suborder of B, M thinks that τ is a Q-name for some regular

suborder Q of B. By elementarity, Vκ thinks that τ is a Q-name for some regular Q of A. For

P of arbitrary size, let τ be a P-name of size < κ, take some regular θ such that P, τ ∈ Hθ,

and take an elementary M ≺ Hθ with P, τ ∈ M such that |M | = κ and M<κ ⊆ M . It is

easy to see that M ∩ P is a regular suborder of P, and so the above considerations apply to

show that there is some regular Q ⊆ P ∩M ⊆ P of size < κ such that τ is a Q-name.

Therefore, if κ is weakly compact and P is κ-c.c., the collection L of all regular suborders

of P of size < κ witnesses that P is (ω, κ)-nicely layered. If P also forces κ = ℵ1, then this

collection also witnesses that P is (ω, κ)-NLC. To check this, take any Q0 ∈ L, any P-name

τ of size < κ, and α < κ. Let H ⊆ Q0 be generic. Since κ is still weakly compact in V [H],

there is some regular Q1 ⊆ P/H of size < κ in V [H] such that the (P/H)-name associated to

τ is a Q1-name. Let β ≥ max{α, |Q0 ∗ Q1|}. Since P/(Q0 ∗ Q1) adds a generic for Col(ω, β),

we have Q2 ∈ L extending Q0 ∗ Q1 such that Q2 ∼ Col(ω, β).

In particular, if κ is weakly compact, then Col(ω,< κ) ∗ Q, where Q is forced to be c.c.c., is

(ω, κ)-NLC. Thus an extremely wide variety of forcing extensions with very different theories

24

can be obtained, each sharing the same reals and same cardinals.

2.1.2 An unfortunate reality

Despite the universality of A(µ, κ), it is difficult to characterize its combinatorial structure.

While it absorbs all of the small sets added by a (µ, κ)-NLC forcing, no such forcing com-

pletely embeds into it. The reader may opt to skip this section, as later results will not

depend on it.

To show this, we first isolate two properties of a forcing extension that depend on two regular

cardinals µ < κ. The author is grateful to Mohammad Golshani for bringing these properties

to his attention.

(1) Levy(µ, κ): (∃A ∈ [κ]κ)(∀y ∈ [κ]µ ∩ V )y * A.

(2) Silver(µ, κ): (∃A ∈ [κ]κ)(∀X ∈ [κ]κ ∩ V )(∃y ∈ [X]µ ∩ V )y ∩ A = ∅.

Note that these are both Σ1 properties of the parameters ([κ]µ)V and ([κ]κ)V . For any partial

order P, and collection of dense subsets D ⊆ P(P) the statement, “There is a filter G ⊆ P

that is D-generic,” is also a Σ1 property of P and D. Now the following proposition either

holds or fails for a given partial order P and cardinals µ < κ:

(∗)µ,κ : (∀X ∈ [P]κ)(∃y ∈ [X]µ)y has a lower bound in P.

Lemma 2.11. If P is a separative partial order that satisfies (∗)µ,κ, preserves the regularity

of κ, and such that d(P � p) = κ for all p ∈ P, then P forces Silver(µ, κ).

Proof. Let {pα : α < κ} be a dense subset of P. Inductively build a dense D ⊆ {pα : α < κ},

putting pα ∈ D just in case there is no β < α such that pβ ∈ D and pβ ≤ pα. D has the

25

property that for all p ∈ D, |{q ∈ D : p ≤ q}| < κ. Fixing a bijection f : D → κ, we claim

that if G ⊆ P is generic, A = f [G] witnesses Silver(µ, κ). Note that since P is nowhere

< κ-dense, A is an unbounded subset of κ. Now let p ∈ D and X = {qα : α < κ} ∈ [D]κ be

arbitrary. There is some B ∈ [κ]κ such that for all α ∈ B, p � qα. For each α ∈ B, choose

rα ≤ p such that rα ⊥ qα. By (∗), there is some y ∈ [B]µ such that {rα : α ∈ y} has a lower

bound r. We have r {qα : α ∈ y} ∩ G = ∅. As p and X were arbitrary, Silver(µ, κ) is

forced.

Lemma 2.12. If P is a κ-c.c. separative partial order of size κ satisfying ¬(∗)µ,κ, then some

p ∈ P forces Levy(µ, κ).

Proof. Suppose X ∈ [P]κ witnesses ¬(∗)µ,κ. By the κ-c.c., there is some p such that p

|X ∩ G| = κ. If y ∈ [X]µ, then 1 y * G, since otherwise some q is a lower bound to y.

Hence p forces that X ∩G witnesses Levy(µ, κ).

Lemma 2.13. Suppose µ < κ, µ is regular for all α < κ, αµ < κ. There are two (µ, κ)-NLC

partial orders P0 and P1 such that P0 forces Levy ∧ ¬Silver, and P1 forces ¬Levy ∧ Silver.

Proof. Let P0 be the Levy collapse Col(µ,< κ) =∏<µ−supp

α<κ Col(µ, α), and let P1 =∏µ−suppα<κ Col(µ, α). It is easy to see that P1 satisfies (∗)µ,κ, while P0 fails this property,

as witnessed by X = P0. Hence by the previous lemmas, P0 forces Levy(µ, κ), and P1 forces

Silver(µ, κ). We must show that the respective negations are also forced.

Let A be a P0-name such that 1 A ∈ [κ]κ. Let p ∈ P be arbitrary, and let γ < κ

be such that supp(p) ⊆ γ. Let X0 = {α < κ : p 1 α /∈ A}. For each α ∈ X0, pick

some qα ≤ p such that qα α ∈ A. By a delta-system argument, let X1 ∈ [X0]κ be

such that there is r ≤ p such that for all α ∈ X1, qα � γ = r, and for α 6= β in X1,

(supp(qα) \ γ) ∩ (supp(qβ) \ γ) = ∅. For any q ≤ r and y ∈ [X1]µ, q 1 y ∩ A = ∅. This

is because for such q, there is some α ∈ y such that (supp(qα) \ γ) ∩ supp(q) = ∅, so q is

26

compatible with qα. Hence r (∃X ∈ [κ]κ ∩ V )(∀y ∈ [X]µ ∩ V )y ∩ A 6= ∅. As A and p were

arbitrary, ¬Silver(µ, κ) is forced.

Now let A be a P1-name such that 1 A ∈ [κ]κ, and let p ∈ P1 be arbitrary. Form X0,

{qα : α ∈ X0}, and X1 like above. We can take a y ∈ [X1]µ such that⋃α∈y qα = q ∈ P1.

Then q y ⊆ A, so q forces ¬Levy(µ, κ).

Corollary 2.14. Suppose µ, κ, P0, and P1 are as above. Let G be P0-generic and H be

P1-generic over V . Let Q ∈ V be partial order. If Q forces Levy(µ, κ), then V [H] has no

Q-generic, and if Q forces Silver(µ, κ), then V [G] has no Q-generic. If Q is κ-c.c. and of

size κ, then no κ-closed forcing extension of V [G] or V [H] can introduce a generic for Q.

Proof. Since V [H] satisfies ¬Levy, and Levy is a Σ1 property with parameters in V , no inner

model of V [H] containing V can satisfy Levy. Likewise, no inner model of V [G] containing

V can satisfy Silver. To see that the non-existence of Q-generics is preserved by κ-closed

forcing, suppose that for some such forcing R ∈ V [G], r V [G]R K is Q-generic over V . Since

Q has size κ, we can build a descending sequence {rα : α < κ} below r such that for all

q ∈ Q, there is rα deciding whether q ∈ K. Let K ′ = {q : (∃α < κ)rα q ∈ K}. Any

maximal antichain A ∈ V contained in Q has size < κ, thus some rα completely decides

A ∩K. Since rα A ∩ K 6= ∅, we must have K ′ ∩ A 6= ∅, so K ′ is Q-generic over V . The

argument for κ-closed forcing over V [H] is the same.

Theorem 2.15. Suppose µ < κ are regular and αµ < κ for all α < κ. No (µ, κ)-NLC forcing

regularly embeds into A(µ, κ). Further, a generic extension by A(µ, κ) has no generic filters

for any κ-c.c. forcing Q such that d(Q � q) ≥ κ for all q ∈ Q.

Proof. First note that we only need to consider Q such that d(Q � q) = κ for all q ∈ Q. For

if p ∈ A(µ, κ) is such that p K is Q-generic, then there would be some q ∈ B(Q) and some

p′ ≤ p such that B(Q) � q completely embeds into A(µ, κ) � p′. Since d(A(µ, κ) = κ, this

implies B(Q) � q ≤ κ.

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Let Q be any κ-c.c. forcing such that d(Q � q) = κ for all q ∈ Q. For any p ∈ Q, if (∗) holds

for Q � p, then p Silver, and otherwise for some q ≤ p, q Levy. Thus Q Levy∨Silver.

Suppose K is Q-generic over V , and X is A(µ, κ)-generic over V . There are two further

forcings R0,R1 over V [X] that respectively get filters G,H such that V [G][X] is P0∗Add(κ)-

generic, and V [H][X] is P1 ∗ Add(κ)-generic. If V [K] |= Levy, then K 6∈ V [H][X], and if

V [K] |= Silver, then K /∈ V [G][X]. Thus V [X] has no Q-generics.

2.2 Construction of a dense ideal

First we will define a useful strengthening of “nicely layered.”

Definition. P is (µ, κ)-very nicely layered (with collapses) when there is a sequence 〈Qα :

α < κ〉 = L such that:

(1) L witnesses that P is (µ, κ)-nicely layered (with collapses),

(2) L is ⊆-increasing,

(3) every subset of P of size < µ with a lower bound has an infimum, and

(4) there is a system of continuous projection maps πα : P → Qα such that for each α,

πα � Qα = id, and for β < α < κ, πβ = πβ ◦ πα.

A typical example is the Levy collapse Col(µ,< κ). In the general case, we will usually

abbreviate the action of the projection maps πα(q) by q � α. In applying clause (3), we will

use the next proposition, proof of which is left to the reader.

Proposition 2.16. If P is a partial order such that every descending chain of length < µ

has an infimum, then every directed subset of size < µ has an infimum.

28

Theorem 2.17. Assume κ carries an almost-huge tower of height δ, and let j : V →M be

given by the tower. Let µ < κ be regular, and let κ ≤ λ < δ. Let X be A(µ, κ)-generic, and

suppose P ∈ V [X] is a partial order such that:

(1) 〈Qα : α < δ〉 witnesses that P is (κ, δ)-very nicely layered, and

(2) for unboundedly many α < δ, |Qα| ≥ |α| and Qα |Qα| = λ.

If H is P-generic over V [X], then in V [X][H], there is a normal, κ-complete, λ-dense ideal

on Pκ(λ).

Proof. Let HX be the A(µ, κ)-generic filter computed from X. Let K ×C be B(µ, κ)/HX ×

Col(µ, λ)-generic over V [X][H], and for brevity letW = V [X][H][K][C]. Note that V [X][K] =

V [G][X], where G∗X is some Col(µ,<κ)∗Add(κ)-generic filter over V . By the distributivity

of B(µ, κ)/HX in V [X], P and its layers Qα are still κ-closed in V [G][X]. For α < β, the

relation Qα “Qβ/Qα is κ-closed” holds in V [G][X] because in V [X], B(µ, κ)/HX × Qα is

κ-distributive. Furthermore, since no sequences of length < µ are added, the forcing given

by the definition of Col(µ, λ) is the same between V , W , and intermediate models.

The forcing to get from V [G] to W is equivalent to (Add(κ)× Col(µ, λ)) ∗ P. Let L be the

collection of subforcings of the form (Add(κ) × Col(µ, λ)) ∗ Qα for α < δ. This sequence

then witnesses the (µ, δ)-NLC property in V [G]. The closure properties are evident, and

since the whole forcing has the δ-c.c., functions from µ to ordinals are indeed captured by

these factors. The “with collapses” part of the definition holds because of clause (2) of the

hypothesis.

Let P0 = P(µ)W , and consider the submodel M(P0). In W , Q0 = P(Add(δ))M(P0) has

cardinality δ. To show this, let Y ⊆ δ be Add(δ)-generic over W . By Theorem 2.10, Y is

A(µ, δ)-generic over V , and hence over M since (Col(µ,< δ) ∗ Add(δ))M = (Col(µ,< δ) ∗

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Add(δ))V by the closure of M . Since M [Y ] thinks j(δ) is inaccessible, M [Y ] |= |Q0| < j(δ),

so W [Y ] |= |Q0| = δ since j(δ) < (δ+)V . Since W |= 2µ = δ, W and W [Y ] have the same

cardinals, so W |= |Q0| = δ. Therefore, working in W , we can inductively build a set X ⊆ δ

that is Add(δ)-generic over M(P0) with X∩κ = X. By Lemma 2.9, X is A(µ, δ)-generic over

M [G]. A further forcing produces G′ ⊇ G, such that G′ ∗ X is Col(µ,<δ) ∗ Add(δ)-generic

over M , so have an elementary j : V [G][X] → M [G′][X] extending j. By elementarity,

for the corresponding filters HX and HX on the respective algebras A(µ, κ)V and A(µ, δ)M ,

we have j[HX ] ⊆ HX . Hence we can define in W the restricted elementary embedding

j : V [X]→M [X].

Now we wish to extend j to have domain V [X][H]. As in the argument for Lemma 2.8,

every element of (Ordµ)W is coded by some element of M and some y ⊆ µ coded in X, so

M [X] is closed under < δ sequences from W . Consequently, H ∩ Qα and j[H ∩ Qα] are in

M [X] for all α < δ. Also, M [X] � “j(P) is (δ, j(δ))-very nicely layered.” Each j[H ∩Qα] is

a directed set of size µ in M [X], so it has an infimum mα ∈ j(Qα).

Let 〈Aα : α < δ〉 ∈ W enumerate the maximal antichains of j(P) from M [X]. (There are

only δ many because M [X] thinks this partial order has inaccessible size j(δ) and is j(δ)-

c.c.) Inductively define an increasing sequence of ordinals 〈αi〉i<δ ⊆ δ, and a corresponding

decreasing sequence of conditions 〈pi〉i<δ ⊆ j(P) as follows.

Assume as the induction hypothesis that we have defined the sequences up to i, and for all

ξ < i and all α < δ, pξ is compatible with mα, and for all ξ < i, there is some a ∈ Aξ such that

pξ ≤ a. Let qi = infξ<i pξ. This is compatible with all mα because for all α, 〈qξ ∧mα : ξ < i〉

is a descending chain in j(P). Let αi ≥ supξ<i αξ be such that Ai ⊆ j(Qαi) and qi ∈ j(Qαi).

This is possible by the chain condition and because j[δ] is cofinal in j(δ). Choose pi ∈ j(Qαi)

below qi ∧ mαi and some a ∈ Ai. pi is compatible with all mα, because for any α > αi,

mα � j(αi) = mαi . This is because for any β < α < δ, H ∩Qβ = {p � β : p ∈ H ∩Qα}, and

the projections are continuous.

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The upward closure of the sequence 〈pi〉i<δ is a filter H which is j(P)-generic over M [X].

For all p ∈ H, j(p) ∈ H since there is some mα ≤ j(p). Thus we get an extended elementary

embedding j : V [X][H]→M [X][H]. In W , we define an ultrafilter U over (P(Pκ λ))V [X][H]:

let A ∈ U iff j[λ] ∈ j(A). Note that j[λ] ∈ Pj(κ)(j(λ))M [X][H]. U is κ-complete and normal

with respect to functions in V [X][H]. If f : Pκ(λ) → λ is a regressive function in V [X][H]

on a set A ∈ U , then j(f)(j[λ]) = j(α) for some α < λ, so {z ∈ A : f(z) = α} ∈ U .

Now the forcing to obtain U was Q = B(µ, κ)/HX ×Col(µ, λ), the product of a κ-dense and

a λ-dense partial order. In V [X][H], let e : P(Pκ λ)→ B(Q) be defined by e(A) = ||A ∈ U ||.

Let I be the kernel of e. I is clearly a normal, κ-complete ideal. e lifts to a boolean embedding

of P(Pκ λ)/I into B(Q). Since Q is λ+-c.c., I is λ+-saturated. If 〈[Aα] : α < λ〉 is a maximal

antichain in Pκ(λ)/I, then ∇Aα is the least upper bound and is in the dual filter to I.

e(∇Aα) = ||∇Aα ∈ U || = 1, and this is the least upper bound in B(Q) to {e(Aα) : α < λ}.

This is because if there were a generic extension in which all Aα /∈ U , then ∇Aα /∈ U as

well since U is normal with respect to sequences from V [X][H]. Therefore e is a complete

embedding, and thus I is λ-dense.

When λ is regular, it is easy to find a forcing P ∈ V [X] satisfying the hypotheses–for example

Col(λ,<δ). Though we do not explicitly assume λ is regular, it is actually required for the

argument. Applying the theorem to the case of singular λ would require, at minimum,

turning an inaccessible into the successor of a singular cardinal with a countably closed

forcing. This was recently observed to be impossible by Asaf Karagila and Yair Hayut, who

communicated their argument to the author in private correspondence.

Theorem 2.18 (Karagila-Hayut). If δ > κ = λ+ and P is a forcing preserving all stationary

subsets of κ ∩ cof(ω), then P cannot force that λ is singular and δ = λ+.

Proof. Let G ⊆ P be generic. If λ is singular and δ is its successor in V [G], then for some

µ < λ, cf(κ) = µ. In V , choose a collection {Sα : α < λ+} of disjoint stationary subsets of

31

κ ∩ cof(ω). In V [G] choose a club {βi : i < µ} ⊆ κ. For each i < µ, there is at most one

α < κ such that βi ∈ Sα, and for each α < κ, there is an i < µ such that βi ∈ Sα. Thus

there is a surjection f : µ→ κ in V [G], and λ is not a cardinal in V [G], a contradiction.

We can also characterize the exact structure of P(Pκ λ)/I, for the ideals produced via

Theorem 2.17. First note the following about the ground model embedding j : V →M . M

is the direct limit of the coherent system of α-supercompactness embeddings jα : V → Mα

for α < δ. Every member of Mα is represented as jα(f)(jα[α]) for some function f ∈ V with

domain Pκ(α). If kα : Mα → M is the factor map such that j = kα ◦ jα, then the critical

point of kα is above α, so kα(x) = kα[x] when Mα � |x| ≤ |α|. Since M is the direct limit,

for any x ∈M , there is some α < δ and some f ∈ V such that

x = kα([f ]) = kα(jα(f)(jα[α])) = j(f)(kα(jα[α])) = j(f)(j[α])).

Let U ⊆ P(Pκ λ)/I be generic over V [X][H], and let jU : V → N be the generic ultrapower

embedding. Since e : P(Pκ λ)/I → B(Q) is a complete embedding, forcing with B(Q)/e[U ]

over V [X][H][U ] produces a model W as above. Notice that the definition of e and U makes

A ∈ U iff j[λ] ∈ j(A). Hence we can define an elementary embedding k : N →M [X][H] by

k([f ]) = j(f)(j[λ]), and we have j = k ◦ jU .

What is the critical point of k? Since N � µ+ = δ, certainly it must be at least δ. Let

β be any ordinal. There is some α such that λ ≤ α < δ and some f ∈ V such that

β = j(f)(j[α]). Let b : λ → α be a bijection in V [X][H]. Then β = j(f)(j(b)[j[λ]]).

Furthermore, j[λ] = k(jU [λ]). Therefore, β = k(jU(f)(jU(b)[jU [λ]])). Thus β ∈ ran(k), and

so k does not have a critical point. Therefore, N = M [X][H]. By the closure of M [X][H],

the generic K × C for Q is in M [X][H] = N ⊆ V [X][H][U ]. So the quotient B(Q)/e[U ] is

trivial and P(Pκ λ)/I ∼= B(Q) � q for some q.

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2.2.1 Minimal generic supercompactness

Generalizing supercompactness, we will say cardinal κ is generically supercompact when for

every λ ≥ κ, there is a forcing P such that whenever G ⊆ P is generic, there is an elementary

embedding j : V → M , where M is a transitive class in V [G], crit(j) = κ, j(κ) > λ, and

Mλ ∩ V [G] ⊆ M . We note that unlike in the case of non-generic supercompactness, the

condition that j[λ] ∈ M does not imply that M is closed under λ-sequences from V [G].

Whenever a supercompact κ is turned into a successor cardinal by a κ-c.c. forcing, we’ll

have that for all λ ≥ κ, there is a normal, fine, precipitous ideal on Pκ(λ) whose generic

embeddings always extend the original supercompactness embedding. But if j : V →M is an

embedding coming from a normal ultrafilter on Pκ(λ), then 2λ<κ< j(κ) < (2λ

<κ)+. If κ = µ+

in a generic extension V [H], and a further extension gives j : V [H] → M [H] ⊆ V [H][G]

extending j, then M [H] is not closed under λ-sequences from V [H][G]. This is because

|λ| = |j(κ)| = µ in V [H][G], while M [H] thinks j(κ) is a cardinal.

Stronger properties of ideals on Pκ(λ) are needed to give genuine generic supercompactness.

One such property is λ+-saturation, which is implied by λ-density. We now sketch how to

get a model in which there is a successor cardinal κ such that for all regular λ ≥ κ, there is a

normal, κ-complete, λ-dense ideal on Pκ(λ). Start with a super-almost-huge cardinal κ and

a regular µ < κ. The first part of the forcing is A(µ, κ). Then we do a proper class iteration,

which we prefer to describe instead as an iteration up to an inaccessible δ > κ such that

Vδ � κ is super-almost-huge.

Let T = {α < δ : κ carries an almost-huge tower of height α}. Let C be the closure of

T , and let 〈αβ〉β<δ be its continuous increasing enumeration. Over V A(µ,κ), let Pδ be the

Easton-support limit of the following:

• Let P0 = Col(κ,< α0).

33

• If β is zero or a successor ordinal, let Pβ+1 = Pβ ∗ Col(αβ, < αβ+1).

• If β is a limit ordinal such that αβ is singular, let Pβ+1 = Col(α+β , < αβ+1).

• If β is a limit ordinal such that αβ is regular, let Pβ+1 = Col(αβ, < αβ+1).

It is routine to verify that this iteration preserves the regularity of the members of T , the

successors of the singular limit points of T , and the regular limit points of T . Further, the

set of non-limit-points of T becomes the set of successors of regular cardinals between κ and

δ.

Let X ⊆ κ be A(µ, κ)-generic over V , and let H ⊆ Pδ be generic over V [X]. Suppose

κ ≤ λ < δ, and λ is regular in V [X][H]. Then there is some successor ordinal β < δ such that

αβ ∈ T and αβ = λ+. Consider the subforcing A(µ, κ)∗Pβ = (A(µ, κ)∗Pβ−1)∗Col(λ,< αβ).

The forcing Pβ is (κ, αβ)-very nicely layered in V [X].

If j : V → Mβ is an almost-huge embedding with critical point κ and j(κ) = αβ, then by

Theorem 2.17, there is a normal, κ-complete, λ-dense ideal on Pκ(λ) in V [X][Hβ]. Now note

that the tail-end forcing Pβ,δ is αβ-closed. Since λ<κ = λ in V [X][Hβ], no new subsets of

Pκ(λ) are added by the tail. The collection {Aα : α < λ} witnessing the λ-density of I retains

this property, as this is a local property of the boolean algebra Pκ(λ)/I and {Aα : α < λ}.

Normality and completeness of I are likewise preserved.

Because of the generality of the hypotheses of Theorem 2.17, this method is quite flexible.

It can done by iterating collapsing posets other than the Levy collapse, or by using products

rather than iterations.

34

2.2.2 Dense ideals on successive cardinals?

At the time of this writing, it is unknown whether there can exist simultaneously a normal

κ-dense ideal on κ and a normal κ+-dense ideal on κ+. The following is the current best

approximation.

Suppose 〈κn : n < ω〉 is a sequence of cardinals such that for all n, κn carries an almost-

huge tower of height κn+1. Such a sequence will be called an almost-huge chain. Obviously,

extending this to sequences of length longer than ω requires an extra idea; perhaps we just

stack one ω-chain above another, or maybe postulate some relationship between the ω-chains.

By Theorem 2.4, such chains occur quite often below a huge cardinal.

Suppose 〈κn : 0 < n < ω〉 is an almost-huge chain, and µ < κ1 is regular. Consider

the full-support iteration P of 〈Pn : n < ω〉, where P0 = A(µ, κ1), and for all n < ω,

Pn+1 = Pn∗A(κn, κn+1). The stage P1 = A(µ, κ1)∗A(κ1, κ2) regularly embeds into A(µ, κ1)∗

(Col(κ1, <κ2) ∗ Add(κ2)). The first two stages here add a normal κ1-dense ideal on κ1 and

make κ1 = µ+, κ2 = µ++. The third stage preserves this since it adds no subsets of κ1.

By Lemma 2.8, the quotient forcing Q to get from V P1 to this three-stage extension is κ2-

distributive. Now the tail-end forcing P/P1 is κ2-strategically closed. Since Q does not add

any plays of the relevant game of length <κ2, P/P1 remains κ2-strategically closed in V P1∗Q,

so forcing with it preserves the κ1-dense ideal on κ1. Also, Q remains κ2-distributive in V P,

since Q × (P/P1) is κ2-distributive in V P1 . It thus remains the case in V P that there is a

κ2-distributive forcing adding a normal κ1-dense ideal on κ1.

Similarly, consider V Pn for n > 1. Pn = Pn−2 ∗ (A(κn−1, κn) ∗ A(κn, κn+1)). Since |Pn−2| =

κn−1 (or µ for n = 2), κn retains an almost-huge tower of height κn+1 in V Pn−2 . Thus

the same argument applies: In V Pn , there is a κn+1-distributive forcing adding a normal

κn-dense ideal on κn, and this remains true in V P. Therefore, we obtain a model in which

for all n > 0, there is a µ+n+1-distributive forcing adding a normal µ+n-dense ideal on µ+n.

35

By repeating this with a tall enough stack of almost-huge chains, we obtain the consistency

of ZFC with the statement, “For all regular cardinals κ, there is a κ++-distributive forcing

adding a normal κ+-dense ideal on κ+.”

36

Chapter 3

Structural constraints

Saturated ideals have a strong influence over the combinatorial structure of the universe

in their vicinity. Phenomena of this type may also be viewed as the universe imposing

constraints on the structural properties of ideals. Below are some of the most interesting

known results to this effect. Proofs can be found in [13].

(1) (Tarski) If I is a nowhere-prime ideal which is κ-complete and µ-saturated for some

µ < κ, then 2<µ ≥ κ.

(2) (Jech-Prikry) If κ = µ+, 2µ = κ, and there is a κ-complete, κ+-saturated ideal on κ,

then 2κ = κ+.

(3) (Jech-Prikry) If κ = µ+, and there is a κ-complete, κ+-saturated ideal on κ, then there

are no κ-Kurepa trees.

(4) (Woodin) If there is a countably complete, ω1-dense ideal on ω1, then there is a Suslin

tree.

(5) (Woodin) If there is a countably complete, uniform, ω1-dense ideal on ω2, then 2ω = ω1.

(Uniform means that all sets of size < ω2 are in the ideal–equivalent to fineness.)

37

(6) (Shelah) If 2ω < 2ω1 , then NSω1 is not ω1-dense.

(7) (Gitik-Shelah) If I is a κ-complete, nowhere-prime ideal, then d(I) ≥ κ.

We note that result (2) easily generalizes to the following: If κ = µ+, 2µ = κ, and there is a

normal, fine, κ-complete, λ+-saturated ideal on Pκ(λ), then 2λ = λ+.

If no requirements are made for the ideal I and the set Z on which it lives, almost no

structural constraints on quotient algebras remain. The following strengthens a folklore

result, probably known to Sikorski. The argument was supplied by Don Monk in personal

correspondence.

Proposition 3.1. Let B be a complete boolean algebra, and let κ be a cardinal such that

2κ ≥ |B|. There is a uniform ideal I on κ such that B ∼= P(κ)/I.

Proof. Let κ, B be as hypothesized. By the theorem of Fichtenholz-Kantorovich and Haus-

dorff (see [21], Lemma 7.7), there exists a family F of 2κ many subsets of κ such that for

any x1, ..., xn, y1, ..., ym ∈ F , x1 ∩ ... ∩ xn ∩ (κ \ y1) ∩ ... ∩ (κ \ ym) has size κ. F generates

a free algebra: closing F under finitary set operations gives a family of sets G such that

any equation holding between elements of G expressed as boolean combinations of elements

of F holds in all boolean algebras. If we pick any surjection h0 : F → B and extend it to

h1 : G→ B in the obvious way, then h1 will be a well-defined homomorphism.

Let Ibd be the ideal of bounded subsets of κ. Since all elements of G are either empty or have

cardinality κ, G ∼= G/Ibd, so h1 has an extension h2 from the algebra generated by G ∪ Ibd

to B, where h2(x) = 0 for all x ∈ Ibd. Finally, by Sikorski’s extension theorem, there is a

further extension to a homomorphism h3 : P(κ) → B. The kernel of h3 is an ideal I such

that P(κ)/I ∼= B.

38

3.1 Cardinal arithmetic and ideal structure

A careful examination of the proof of Woodin’s theorem (5) shows that ω2 can be replaced by

any ωn, 2 ≤ n < ω. Aside from that, Woodin’s argument is rather specific to the cardinals

involved. In [13], Foreman asked (Open Question 27) whether the analogous statement holds

one level up:

Question (Foreman). Does the existence of an ω2-complete, ω2-dense, uniform ideal on ω3

imply that 2ω1 = ω2?

To answer this, we invoke an easy preservation lemma about ideals under small forcing. If I

is an ideal, P is a partial order, and G ⊆ P is generic, then I denotes the ideal generated by

I in V [G], i.e. {X : (∃Y ∈ I)X ⊆ Y }.

Lemma 3.2. Suppose I is a κ-complete ideal on Z ⊆ P(X), P is partial order, and G is

P-generic.

(1) If sat(P) ≤ κ, then I is κ-complete in V [G].

(2) If d(P) < κ, then d(I)V [G] ≤ d(I)V .

Proof. For (1), let s be a P-name for a sequence of elements of I of length less than κ. By

κ-saturation, let β < κ be such that 1 dom(s) ≤ β. For each α < β, let Aα be a maximal

antichain such that for p ∈ Aα, p s(α) ⊆ bpα, where bpα ∈ I. Then B =⋃p,α b

pα ∈ I, and

1 ⋃s ⊆ B.

For (2), let D ⊆ P be a dense set of size less than κ, and let A ∈ I+. Then A =⋃d∈D∩G{z :

d z ∈ A}. By (1), there is some d ∈ D such that {z : d z ∈ A} /∈ I. This shows that

(P(Z)/I)V is dense in (P(Z)/I)V [G], and the conclusion follows.

Corollary 3.3. If there is a κ+-complete, κ+-dense, uniform ideal on κ++, then 2κ = κ+.

39

Proof. Suppose for a contradiction that f : P(κ)→ κ++ is a surjection. Let P = Col(ω, κ),

and let G be P-generic. Since d(P) = κ, Lemma 3.2 implies that I is κ+-complete and

κ+-dense in V [G]. Furthermore, in V [G], κ+ = ω1 and κ++ = ω2. Thus Woodin’s theorem

implies that V [G] � CH. However, f witnesses the failure of CH, a contradiction.

Another interesting constraint can be derived from the following:

Theorem 3.4 (Shelah [30]). Suppose V ⊆ W are models of ZFC. If κ is a regular cardinal

in V , and cf(κ) 6= cf(|κ|) in W , then (κ+)V is not a cardinal in W .

Corollary 3.5. If κ = µ+, λ ≥ κ is regular, and I is a normal, fine, κ-complete, λ+-

saturated ideal on Pκ(λ), then {z : cf(z) = cf(µ)} ∈ I∗.

Proof. Let G be a generic ultrafilter extending I∗. Since crit(j) = κ and λ+ is preserved,

j(κ) = λ+, and |λ| = µ in V [G]. By Shelah’s theorem, cf(λ) = cf(µ) in V [G] and in

the ultrapower M since Mµ ∩ V [G] ⊆ M . Since 1 [id] = j[λ], Los’s theorem gives

{z : cf(z) = cf(µ)} ∈ I∗.

Theorem 3.6. Suppose κ = µ+, and I is a normal, fine, κ+-saturated ideal on κ. Then

P(κ)/I is cf(µ)-distributive iff µ<cf(µ) = µ.

Proof. Suppose P(κ)/I is cf(µ)-distributive, and let {fα : α < δ} be an enumeration of

[µ]<cf(µ), where δ is a cardinal. If µ < δ, then for any P(κ)/I-generic G, ([µ]<cf(µ))V is

a proper subset of ([µ]<cf(µ))V [G], since j[δ] 6= j(δ). This contradicts the distributivity of

P(κ)/I.

Since P(κ)/I is κ+-saturated, it is cf(µ)-distributive iff it is (cf(µ), κ)-

distributive. Let G be P(κ)/I-generic and let M be the generic ultrapower. Let β < cf(µ),

and suppose f ∈ V [G] is a function from β to κ. By Theorem 1.16, f ∈M . By Corollary 3.5,

M � cf(κ) = cf([id]) = cf(µ). Thus there is a γ < κ such that ran(f) ⊆ γ. Observe that

j(βγ) = (βγ)M = (βγ)V , since µβ < κ. Hence f ∈ V .

40

3.2 Stationary reflection

A stationary subset S of a regular cardinal κ is said to reflect if there is some α < κ such that

S ∩ α is stationary in α. A collection of stationary subsets {Si : i < δ} of κ is said to reflect

simultaneously if there is some α < κ if Si ∩ α is stationary for all i < δ. It is well known

that if κ = µ+ and X is a set of regular cardinals below µ, then the statement that every

stationary subset of {α < κ : cf(α) ∈ X} reflects contradicts �µ, and the statement that

every pair of stationary subsets of {α < κ : cf(α) ∈ X} reflect simultaneously contradicts

the weaker principle �(κ).

Theorem 3.7. Suppose there is a κ+-complete, κ++-saturated, uniform ideal on κ+n for

some n ≥ 2. Then for 2 ≤ m ≤ n, every collection of κ many stationary subsets of κ+m

contained in cof(≤ κ) reflects simultaneously.

Proof. Suppose I is such an ideal and j : V → M ⊆ V [G] is a generic embedding arising

from the ideal. The critical point of j is κ+, and all cardinals above κ+ are preserved. Since

I is uniform, and there is a family of κ+n+1 many almost-disjoint functions from κ+n to

κ+n, j(κ+n) ≥ (κ+n+1)V . The first n − 1 cardinals in V above κ must map onto the first

n − 1 cardinals in M above κ. But in M , there are at least n − 1 cardinals in the interval

(κ, (κ+n+1)V ) since all cardinals above κ+ are preserved. Thus if j(κ+n) > (κ+n+1)V , then

κ+n+1 would be collapsed. So for 1 ≤ m ≤ n, j(κ+m) = (κ+m+1)V .

Let {Sα : α < κ} be stationary subsets of κ+m concentrating on cof(≤ κ), where 2 ≤ m ≤ n.

By the κ++-chain condition, these sets remain stationary in V [G]. By the above remarks,

γ = sup(j[κ+m]) < j(κ+m). For each α, j � Sα is continuous since κ < crit(j). For each α, let

Cα be the closure of Sα. In V [G], we can define a continuous increasing function f : Cα → γ

extending j � Sα by sending sup(Sα ∩ β) to sup(j[Sα ∩ β]) when β is a limit point of Sα.

This shows that j[Sα] is stationary in γ. Now M may not have j[Sα] as an element, but it

satisfies that j(Sα) ∩ γ is stationary in γ. Furthermore, j({Sα : α < κ}) = {j(Sα) : α < κ},

41

and M sees that these all reflect at γ. By elementarity, the Sα have a common reflection

point.

Proposition 3.8. Suppose µ, κ, λ are regular cardinals such that ω < µ < κ = µ+ < λ, and

I is an ideal on Pκ(λ) as in Theorem 2.17. Then every collection {Si : i < µ} of stationary

subsets of λ ∩ cof(ω) reflects simultaneously.

Proof. The algebra P(Pκ(λ))/I is isomorphic B(P × Q), where P is κ-dense and P “Q

is countably closed.” Forcing with P × Q thus preserves the stationarity of any subset of

λ ∩ cof(ω). If j : V → M ⊆ V [G] is a generic embedding arising from the ideal, then since

j[λ] ∈M and M thinks j(λ) is regular, γ = sup(j[λ]) < j(λ). The restriction of j to each Si

is continuous, and as above we may define in V [G] a continuous increasing function from the

closure of Si into γ, showing j[Si] is stationary in γ for each i. Thus M |= (∀i < µ)j(Si)∩ γ

is stationary, so by elementarity, the collection reflects simultaneously.

3.3 Nonregular ultrafilters

The computation of the cardinality of ultrapowers is an old problem of model theory. Orig-

inally, it was conjectured that if µ, κ are infinite cardinals, and U is a countably incomplete

uniform ultrafilter on κ, then |µκ/U | = µκ [6]. It was shown by Donder [9] that this conjec-

ture holds in the core model below a measurable cardinal. A key tool in such computing the

size of ultrapowers is the notion of regularity:

Definition. An ultrafilter U on Z is called (µ, κ)-regular if there is a sequence 〈Aα : α <

κ〉 ⊆ U such that for any Y ⊆ κ of order type µ,⋂α∈Y Aα = ∅.

Theorem 3.9 (Keisler [23]). Suppose U is a (µ, κ)-regular ultrafilter on Z, witnessed by

〈Aα : α < κ〉. For each z ∈ Z, let βz = ot({α : z ∈ Aα}) < µ. Then for any sequence of

ordinals 〈γz : z ∈ Z〉, we have |∏γβzz /U | ≥ |

∏γz/U |κ.

42

Obviously any uniform ultrafilter on a cardinal κ is (κ, κ)-regular. Also, any fine ultrafilter

on Pκ(λ) is (κ, λ)-regular, as witnessed by 〈α : α < λ〉. Much can be seen by exploiting a

connection between dense ideals and nonregular ultrafilters.

Lemma 3.10 (Huberich [18]). Suppose B is a complete boolean algebra of density κ, where

κ is regular. Then there is an ultrafilter U on B such that whenever X ⊆ B and∑X ∈ U ,

then there is Y ⊆ X such that |Y | < κ and∑Y ∈ U .

Proof. Let D = {dα : α < κ} be dense in B. For any maximal antichain A ⊆ B, let γA > 0

be least such that for all α < γA, there are β < γA and a ∈ A such that dβ ≤ dα ∧ a. Let

CA = {d ∈ D � γA : (∃a ∈ A)d ≤ a}. Let F = {∑CA : A is a maximal antichain}.

We claim F has the finite intersection property. Let A1, ..., An be maximal antichains. We

may assume γA1 ≤ ... ≤ γAn . Let dα1 ≤ d0 ∧ a1 for some a1 ∈ A1, where α1 < γA1 . Let

dα2 ≤ dα1∧a2 for some a2 ∈ A2, where α2 < γA2 . Proceeding inductively, we get a descending

chain dα1 ≤ ... ≤ dαn , where each dαi ≤ ai for some ai ∈ Ai. Thus dαn ≤∑CA1∧ ...∧

∑CAn .

Let U ⊇ F be any ultrafilter. If∑X ∈ U , then we can find an antichain A that is maximal

below∑X such that (∀a ∈ A)(∃x ∈ X)a ≤ x. Extending A it to a maximal antichain A′,

we have∑CA′ ∈ F . Since |CA′ | < κ, the conclusion follows.

Lemma 3.11. Suppose κ = µ+, λ is regular, and I is a normal and fine, κ-complete, λ-dense

ideal on Z ⊆ Pκ(λ). Then any ultrafilter U ⊇ I∗ given by Lemma 3.10 is (cf(µ) + 1, λ)-

regular.

Proof. By Corollary 3.5, {z : cf(z) = cf(µ)} ∈ I∗. For such z, choose Az ⊆ z of order

type cf(µ) that is cofinal in z. Let U be given by Lemma 3.10. We will inductively build a

sequence of intervals {(xα, yα) : α < λ}, each contained in λ, such that yα < xβ when α < β,

and such that for all α, {z : Az ∩ (xα, yα) 6= ∅} ∈ U .

43

Suppose we have constructed the intervals up to β. Let λ > xβ > sup{yα : α < β}. For

z ∈ xβ, let yβ(z) ∈ z be such that Az ∩ (xβ, yβ(z)) 6= ∅. Since I is normal, there is a maximal

antichain A of I-positive sets such that for all a ∈ A, yz(β) is the same for all z ∈ a. There

is some A′ ⊆ A of size < λ such that∑A′ ∈ U . Let yβ > xβ be such that for z ∈ a ∈ A′,

yβ(z) < yβ.

Now for α < λ, let Xα = {z : Az ∩ (xα, yα) 6= ∅}. Since each Az has ordertype cf(µ) and the

intervals (xα, yα) are disjoint and increasing, each Az cannot have nonempty intersection with

all intervals in some sequence of length greater than cf(µ). Thus if s ⊆ λ and z ∈⋂α∈sXα,

then ot(s) ≤ cf(µ).

Lemma 3.12. Suppose κ < λ are regular, and I is a κ-complete, λ-dense ideal on Z such

that P(Z)/I is complete. Then there is an ultrafilter U ⊇ I∗ such that for all α < κ,

|αZ/U | ≤ 2<λ.

Proof. Suppose α < κ, and let D witness the λ-density of I, and let U ⊇ I∗ be given by

Lemma 3.10. First we count certain special members of αZ/U . Choose an antichain A ⊆ D

of size < λ, and choose f : A → α. There are∑

γ<λ λγ · αγ = 2<λ many choices. Using

κ-completeness, let {Bβ : β < α} be pairwise disjoint and such that each [Bβ]I =∑f−1(β).

Let gf : Z → α be defined by gf (z) = β if z ∈ Bβ and gf (z) = 0 if z /∈⋃β<αBβ.

Now let g : Z → α be arbitrary. By κ-completeness, A = {g−1(β) : β < α and g−1(β) ∈ I+}

forms a maximal antichain. Let A′ ⊆ D be a maximal antichain refining A. There is some

A′′ ⊆ A′ of size < λ such that∑A′′ ∈ U . Let f : A′′ → α be defined by f(a) = β iff

a ≤I g−1(β). If [B]I =∑A′′, then {z ∈ B : g(z) 6= gf (z)} ∈ I, so g =U gf .

The following contrasts with the consistency results of Chapter 2:

Theorem 3.13. Suppose µ is a singular cardinal such that 2cf(µ) < µ, λ is regular, and

2<λ < 2λ. Then there is no normal and fine, λ-dense ideal on Pµ+(λ).

44

Proof. Suppose such an ideal exists, and let U be given by Lemma 3.10. By Lemma 3.12,

|∏µ/U | ≤ 2<λ. But by Lemma 3.11, U is (cf(µ)+1, λ)-regular. Hence, Theorem 3.9 implies

that |∏

2cf(µ)/U | ≥ 2λ, a contradiction.

Corollary 3.14. Suppose µ is a singular cardinal such that 2cf(µ) < µ. Then for all α there

is a regular λ > α such that there is no normal and fine, λ-dense ideal on Pµ+(λ).

Proof. We will show that there is a proper class of regular cardinals λ such that 2<λ < 2λ.

Assume for a contradiction that this fails. Let α be arbitrary, and let κ = 2α. We will show

by induction the impossible conclusion that 2β = κ for all β ≥ α. Suppose that this holds for

all γ < β. If β is regular, then by a general equation shown in [21], 2<β = 2β by assumption,

so 2β = κ. If β is singular, then 2β = (2<β)cf(β) = κcf(β). For regular γ < β above cf(β),

(2γ)cf(β) = κcf(β) = 2γ = κ.

Corollary 3.15. If κ is singular such that 2cf(κ) < κ, then there is no uniform, κ+-complete,

κ+-dense ideal on κ+n for n ≥ 2.

Proof. Assume I is a uniform, κ+-complete, κ+-dense ideal on κ+n for some n ≥ 2. Define

φ : P(κ+)→ P(κ+n)/I by X 7→ ||κ+ ∈ j(X)||P(κ+n)/I . Let J = kerφ. φ lifts to an embedding

of P(κ+)/J into P(κ+n)/I. Since J is clearly normal and κ++-saturated, the embedding is

regular, since for a maximal antichain {Aα : α < κ+}, κ+ ∈ j(∇α<κ+Aα), so it is forced

that for some α < κ+, φ(Aα) is in the generic filter. Thus J is a normal κ+-dense ideal on

κ+. We have 2<κ+< 2κ

+by Corollary 3.3, so κ cannot be singular such that 2cf(κ) < κ.

These methods can also be used to deduce more cardinal arithmetic consequences of dense

ideals. First we need a few more lemmas:

Theorem 3.16 (Kunen-Prikry [26]). If κ is regular and U is a (κ+, κ+)-regular ultrafilter,

then U is (κ, κ)-regular.

45

Lemma 3.17. Suppose (L,<) is a linear order such that for all x ∈ L, |{y ∈ L : y < x}| ≤ κ.

Then |L| ≤ κ+.

Corollary 3.18. Suppose there is a κ+-complete, κ+-dense ideal on κ+n, where n ≥ 2. Then

for 0 ≤ m ≤ n, 2κ+m

= κ+m+1.

Proof. Let I be such an ideal, and let U ⊇ I∗ be given by Lemma 3.10. By Lemma 3.12,

|κκ+n/U | ≤ 2κ, which is κ+ by Corollary 3.3. Note that for any cardinal µ, any ultrafilter

V on a set Z, and any g : Z → µ+, {[f ]V : f <V g} has cardinality at most |µZ/V |. Thus,

applying Lemma 3.17 inductively, we get that |(κ+m)κ+n/U | ≤ κ+m+1 for all m < ω.

U is (κ+n, κ+n)-regular, so by Theorem 3.16, it is (κ+m, κ+m)-regular for m ≤ n. Assume for

induction that 2κ+r

= κ+r+1 for r < m ≤ n; note the base case m = 1 holds. Let {Xα : α <

κ+m} witness (κ+m, κ+m)-regularity, and let βz = ot({α : z ∈ Xα}). By Theorem 3.9 and

the above observations, we have:

2κ+m ≤ |

∏2βz/U | ≤ |

∏2κ

+m−1

/U | = |∏

κ+m/U | ≤ κ+m+1.

We note that if the hypothesis of Corollary 3.18 is consistent, then no cardinal arithmetic

above κ+n can be deduced from it, since any forcing which adds no subsets of κ+n will

preserve the relevant properties of the ideal.

By combining this technique with the results of Chapter 2, we can answer the following,

which was Open Question 16 from [13]:

Question (Foreman). Is it consistent that there is a uniform ultrafilter U on ω3 such that

ωω3/U has cardinality ω3? Is it consistent that there is a uniform ultrafilter U on ℵω+1 such

that ωℵω+1/U has cardinality ℵω+1? Give a characterization of the possible cardinalities of

ultrapowers.

46

Theorem 3.19. Assume ZFC is consistent with a super-almost-huge cardinal. Then it is

consistent that every regular uncountable cardinal κ carries a uniform ultrafilter U such that

|ωκ/U | = κ.

This follows from Chapter 2 and the next result.

Lemma 3.20. Suppose κ = µ+, GCH holds at cardinals ≥ µ, and for all regular λ ≥ κ,

there is a normal and fine, κ-complete, λ-dense ideal on Pκ(λ). Then for every regular λ,

there is a uniform ultrafilter U on λ such that |µλ/U | = µ.

Proof. Let I be a normal and fine, κ-complete, λ-dense ideal on Z = Pκ(λ), where κ = µ+

and λ is regular. Let U ⊇ I∗ be given by Lemma 3.12, so that |µZ/U | ≤ 2<λ. If 2<λ = λ,

then |µZ/U | ≤ λ, and we can assume U is a uniform ultrafilter on λ with the same property.

Since 2µ = κ and any ultrafilter extending I∗ is (κ, λ)-regular, Theorem 3.9 implies that

|κZ/U | > λ, and Lemma 3.17 implies that |κZ/U | ≤ |µZ/U |+. Thus |µZ/U | = λ.

The following extra conclusion can be immediately deduced in the case of µ < ℵω and λ = ρ+,

where cf(ρ) = ω. Suppose µ = ωn. Since |ωZn+1/U | > λ, we cannot have |ωZm/U | < ρ for any

m, since by Lemma 3.17, we would have |ωZr /U | < ρ for all r < ω. Also, U is (ω, ω)-regular,

so Theorem 3.9 implies that |ωZ/U | ≥ |ωZ/U |ω. Thus |ωZm/U | = λ for all m ≤ n.

47

Chapter 4

Ulam’s problem and regularity of

ideals

Taylor [33] generalized the notion of regularity of ultrafilters to arbitrary ideals:

Definition. An ideal I is (µ, κ)-regular if for any sequence 〈Aα : α < κ〉 ⊆ I+, there is a

sequence 〈Bα : α < κ〉 ⊆ I+ such that Bα ⊆ Aα for all α, and for any Y ⊆ κ of order type

µ,⋂α∈Y Bα = ∅.

Note that if I∗ is an ultrafilter, then I is (µ, κ)-regular iff I∗ is (µ, κ)-regular per the definition

in the previous chapter, since if 〈Xα : α < κ〉 witnesses (µ, κ)-regularity in the old sense,

and we are given 〈Aα : α < κ〉 ⊆ I+ = I∗, the we can take Bα = Aα ∩Xα. We will simply

call a κ-complete ideal on κ regular if it is (2, κ)-regular, and a normal and fine ideal on

Z ⊆ P(λ) regular if it is (2, λ)-regular. Taylor [33] proved the following connection between

dense ideals, nonregular ideals, and Ulam’s measure problem:

Theorem 4.1 (Taylor). The following are equivalent:

(1) There is a countably complete ω1-dense ideal on ω1.

48

(2) There is a set {Iα : α < ω1} of normal and fine ideals on ω1 such that every A ⊆ ω1 is

measureable in one of them.

(3) There is a countably complete nonregular ideal on ω1.

We will investigate the extent to which Taylor’s theorem generalizes to ideals on Pκ(λ).

The implications (1) ⇒ (2) and (2) ⇒ (3) will go through in general, but when κ 6= ω1,

the argument for (3) ⇒ (1) will seem to require some additional assumptions. In the next

chapter we will produce models of set theory that show that (3) does not imply either (1)

or (2) when κ 6= ω1.

4.1 Generalizing Taylor’s theorem

The implication (1)⇒ (2) is trivial. If I is a normal and fine, λ-dense ideal on Z, as witnessed

by {Aα : α < λ}, then every subset of Z is measurable in I � Aα for some α < λ. Also, the

implication (1) ⇒ (3) is immediate. If {Aα : α < λ} is dense, and {Bα : α < λ} is a disjoint

refinement into I-positive sets, then there is some Aα ⊆I B0 with α 6= 0, which contradicts

that B0 ∩Bα = ∅. The implication (2) ⇒ (3) requires more work.

Lemma 4.2. Suppose {Iα : α < λ} is a collection of normal and fine ideals on Z ⊆ P(λ),

each of which is nowhere λ+-saturated. Then there is a collection {Xα : α < λ+} of subsets

of Z such that (∀α < λ+)(∀β < λ)Xα ∈ I+β , and for α < β < λ+, Xα ∩Xβ is nonstationary.

Proof. First note that whenever I is normal ideal on Z ⊆ P(λ), and {Aα : α < λ+} is an

antichain, then we can refine the Aα’s so that their pairwise intersections are nonstationary.

For each α < λ+, let fα : λ → α be a surjection. Let Bα = Aα \⋃β<λAfα(β) ∩ β. So when

α 6= β, Bα ∩Bβ ∩ γ = ∅ for some γ.

49

Let {Iα : α < λ} be as hypothesized, and for each Iα pick some antichain {Aαβ : β < λ+}

such that the pairwise intersections are nonstationary. Let h : λ→ λ be defined by h(α) =

the least γ such that {Aγβ : β < λ+} has λ+ many Iα-positive sets. Let δ < λ+ be sup{β :

(∃α < λ)(∃γ < h(α))Aγβ ∈ I+α }. Recursively construct a one-to-one sequence 〈Bα : α < λ〉

such that Bα = Ah(α)β for some β > δ, so Bα ∈ I+

α for all α < λ, and Bα ∩ Bβ ∈ Iα when

h(β) ≤ h(α). Put Cα = Bα ∩ α \⋃h(β)≤h(α) Bβ ∩ β, so Cα ∈ I+

α and Cα ∩ Cβ = ∅ when

α 6= β. Now pick {Dαβ : α < λ, β < λ+} such that each Dα

β is an Iα-positive subset of Cα,

and Dαβ ∩ Dα

γ is nonstationary when β 6= γ. For β < λ+, let Eβ =⋃α<λD

αβ ∩ α. Then

Eβ ∩ Eγ =⋃α<λD

αβ ∩Dα

γ ∩ α, and each Eβ is Iα-positive for all α.

Lemma 4.3. Suppose I is a normal and fine ideal on Z ⊆ P(X). Then I is |X|+-saturated

iff every normal and fine J ⊇ I is equal to I � A for some A ⊆ Z.

Proof. Suppose I is |X|+-saturated. Let {Ax : x ∈ X} be a maximal antichain in J ∩ I+.

Then [∇Ax] is the largest element of P(Z)/I whose elements are in J . Thus J = I �

(Z \ ∇Ax). Now suppose I is not |X|+-saturated, and let {Aα : α < δ} be a maximal

antichain where δ ≥ |X|+. Let J be the ideal generated by⋃{Σα∈Y [Aα] : Y ∈ P |X|+(δ)}.

Then J is a normal, fine, proper ideal extending I. J cannot be equal I � A for some A ∈ I+

because if so, there is some α where A ∩ Aα ∈ I+. A ∩ Aα ∈ J by construction, but every

I-positive subset of A is (I � A)-positive.

Theorem 4.4. If there is a set {Iα : α < λ} of normal and fine ideals on Z ⊆ P(λ) such

that every A ⊆ Z is measurable in one of them, then there is a normal, fine, nonregular ideal

on Z.

Proof. Let J = {Iα : α < λ} be as hypothesized, and assume for a contradiction that every

normal and fine ideal on Z is regular. Let J0 = {I ∈ J : I is nowhere λ+-saturated}, and

for I ∈ J \ J0, choose AI such that I � AI is λ+-saturated, and let J1 = {I � AI : I ∈ J

50

and I is somewhere λ+-saturated}. Clearly, every subset of Z is measurable by some ideal

in J0 ∪ J1.

Now let {Aα : α < λ+} be such that each Aα is I-positive for all I ∈ J0 and Aα ∩ Aβ is

nonstationary for α 6= β. Since at most one Aα can be I-measure one for any I ∈ J1, there is

some α < λ+ such that Z \Aα is I-positive for all I ∈ J1. Let J2 = {I � (Z \Aα) : I ∈ J1}.

Let J =⋂J2, which is clearly normal. Since the J-positive sets are those that are I-positive

for some I ∈ J2, J is λ+-saturated. Thus by Lemma 4.3, each I ∈ J1 is J � BI for some

BI ∈ J+. Since we assume J is regular, there is a collection {CI : I ∈ J2} such each CI is a

J-positive subset of BI , and CI0∩CI1 = ∅ for I0 6= I1. Split each CI into two disjoint positive

sets D0I , D

1I , and let Ei =

⋃I∈J1 D

iI for i < 2. Split Aα into two disjoint sets F0, F1 that are

I-positive for all I ∈ J0. Then E0 ∪ F0, E1 ∪ F1 are nonmeasurable for all I ∈ J .

The argument for (3) ⇒ (1) in the case of ω1 uses a result of Baumgartner, Hajnal, and

Mate [3] about normal ideals on ω1. The following generalizes their argument.

Lemma 4.5. Suppose J is a normal, fine, κ+-complete ideal on Z ⊆ Pκ+(X) which is

nowhere |X|-dense, and P(Z)/J is κ-strategically closed. Then J is regular.

Proof. Let {Mαx : α < κ, x ∈ X} be a generalized Ulam matrix as in Lemma 1.4, so that for

each x ∈ X, x =⋃α<κM

αx , and Mα

x ∩Mαy = ∅ for x 6= y.

Now let {Ax : x ∈ X} be any sequence of J-positive sets. For α < κ, let Aαx = Ax ∩Mαx .

Note that κ+-completeness implies ∀x∃αAαx ∈ J+. Let S = {(α, x) : Aαx ∈ J+}. Using

the winning strategies {σp : p ∈ P(Z)/J} for Even, where σp has opening move p, we will

inductively build descending sequences of sets below Aαx to create a disjoint refinement.

If (α, x) ∈ S, we play a game below Aαx , and denote representatives for Even’s plays by

Aα,βx and those for Odd’s plays immediately following by Bα,βx . Let γ < κ and assume for

51

induction that for (α, x) ∈ S we have chosen sequences 〈Aα,βx 〉β<γ and 〈Bα,βx 〉β<γ satisfying

the following:

(1) Gα,γx = 〈[Aα,0x ], [Bα,0

x ], ..., [Aα,βx ], [Bα,βx ], ...〉β<γ is a sequence played according to σ[Aαx ].

(2) If β1 < β2 < γ, then Aαx ⊇ Aα,β1x ⊇ Bα,β1x ⊇ Aα,β2x .

(3) If β < γ, β 6= α and x, y ∈ X, then Bβ,βx ∩Bα,β

y = ∅.

For each (α, x) ∈ S, let Aα,γx be a representative of σ[Aαx ](Gα,γx ) that is a subset of all previously

chosen representatives. Since P(Z)/J is nowhere |X|-dense, for any x ∈ X, the set:

{[Aγ,γx ∩ Aα,γy ] : (α, y) ∈ S} ∩ J+

is not dense below Aγ,γx . Thus we may choose a J-positive Bγ,γx ⊆ Aγ,γx such that (Aγ,γx ∩

Aα,γy ) \Bγ,γx ∈ J+ for all α, y such that Aγ,γx ∩Aα,γy ∈ J+. For (α, y) ∈ S such that α 6= γ, let

Bα,γy = Aα,γy \

⋃x∈X B

γ,γx .

Note that when (γ, x), (α, y) ∈ S and γ 6= α,

(Aγ,γx ∩ Aα,γy ) \Bγ,γx = (Aγ,γx ∩ Aα,γy ) \

⋃w∈X

Bγ,γw = Aγ,γx ∩Bα,γ

y ,

since Bγ,γw ∩ Aγ,γx = ∅ when x 6= w. If Aγ,γx ∩ Aα,γy ∈ J+ for some x, then Bα,γ

y ∈ J+. If

(α, y) ∈ S and Aγ,γx ∩ Aα,γy ∈ J for all x, then Bα,γy ∈ J+ by the normality of J .

Clearly the induction hypotheses hold with respect to γ + 1. In the end, the collection

{Bα,αx : (α, x) ∈ S} is a pairwise disjoint refinement of {Aαx : (α, x) ∈ S} into J-positive sets.

As {Ax : x ∈ X} was arbitrary, this shows that J is regular.

Corollary 4.6. Let Z ⊆ Pω1(λ) be stationary. The following are equivalent:

(1) There is a normal, fine, λ-dense ideal on Z.

52

(2) There is a set {Iα : α < λ} of normal and fine ideals on Z such that every A ⊆ Z is

measurable in one of them.

(3) There is a normal, fine, nonregular ideal on Z.

Proof. We’ve already seen (1) ⇒ (2) ⇒ (3). To see (3) ⇒ (1), note that every normal and

fine ideal on Z is ω-strategically closed. Thus by Lemma 4.5, any normal, fine, nonregular

ideal on Z is somehwere λ-dense.

The theorem works for ideals on Pκ(λ) for κ = µ+ > ω1, if we also assume that every

λ+-saturated ideal on Pκ(λ) is µ-strategically closed. If we could reduce the hypothesis of

µ-strategic closure in Lemma 4.5 to µ-distributivity, then by Proposition 3.6, we could prove

under µ<µ = µ that for normal ideals on µ+, (1), (2), and (3) are equivalent. But later we

will produce models proving such a reduction is impossible. However, under GCH, we can

prove the equivalence of (1) and (2) by a different argument.

Definition. If B is a boolean algebra, D ⊆ B+ is called weakly dense if (∀b ∈ B+)(∃d ∈

D)(d ≤ b or d ≤ ¬b).

Lemma 4.7. Let Z ⊆ P(λ) be stationary. The following are equivalent.

(1) There is a collection {Iα : α < λ} of normal and fine ideals on Z such that every subset

of Z is measurable in one of them.

(2) There is a λ+-saturated ideal I on Z such that P(Z)/I has a weakly dense subset of size

≤ λ.

Proof. Let J = {Iα : α < λ} be such that every subset of Z is measurable in one of

them. Let J0 = {I ∈ J : I is nowhere λ+-saturated}, and choose sets AI such that

J1 = {I � AI : I ∈ J \ J0 and I � AI is λ+-saturated}. As in the proof of Theorem 4.4,

53

there is some A ⊆ Z such A is positive for all I ∈ J0, Z \ A is positive for all I ∈ J1, and

every subset of Z is measurable by some ideal in J0 ∪ J1. Let J =⋂I∈J1 I � (Z \ A). J is

normal, fine, and λ+-saturated. By Lemma 4.3, for all I ∈ J1, there is some BI ∈ J+ such

that I � (Z \ A) = J � BI .

We claim {BI : I ∈ J1} is weakly dense in P(Z \ A)/J . Otherwise, there is a partition of

Z \ A into two J-positive sets B0, B1 such that for no I ∈ J1 is B0 or B1 in (J � BI)∗. We

can partition A into A0, A1 that are I-positive for all I ∈ J0. Then A0 ∪B0 and A1 ∪B1 are

nonmeasurable for all I ∈ J .

Conversely, if I is an ideal on Z such that P(Z)/I has a weakly dense subset {Aα : α < λ},

then for every B ⊆ Z, there is some α < λ such that B ∈ (I � Aα) ∪ (I � Aα)∗.

Theorem 4.8 (Bozeman [4]). Suppose B is a complete boolean algebra, D ⊆ B is weakly

dense, |D| = κ, and 2<κ = κ. Then there is a b ∈ B and a set E of size κ that is dense below

b.

Corollary 4.9. Suppose 2<λ = λ. Then there is a normal, fine, λ-dense ideal on Z ⊆ P(λ)

iff there is a collection {Iα : α < λ} of normal and fine ideals on Z such that every subset of

Z is measurable in one of them.

Corollary 4.10. Assume GCH, κ is the successor of a singular cardinal, and λ ≥ κ is

regular. If {Iα : α < λ} is a set of normal and fine ideals on Pκ(λ), then there is a set

X ⊆ Pκ(λ) that is not measurable in any them.

4.2 Reduction to normality and degrees of regularity

In many cases, questions about κ-complete ideals on κ reduce to questions about normal and

fine ideals on κ. We use this technique to extend some of the previous results to non-normal

ideals, and then show that most degrees of regularity are equivalent under enough GCH.

54

On the other hand, a model constructed by Shelah shows the importance of the normality

assumption in clause (2) of Taylor’s theorem.

Lemma 4.11 (Baumgartner-Hajnal-Mate [3]). If I is a κ-complete, nowhere κ+-saturated

ideal, then I is regular.

Lemma 4.12 (Taylor [33]). Suppose I is a κ-complete, κ+-saturated ideal on κ = µ+. Then

there is A ∈ I+ and a bijection f : κ→ κ such that f∗(I � A) = {X ⊆ κ : f−1(X) ∈ I � A}

is normal.

Lemma 4.13 (Taylor [33]). Let I be a κ-complete ideal on Z. Suppose that the set {A ∈

I+ : (I � A) is (µ, κ)-regular} is dense in P(Z)/I. Then I is (µ, κ)-regular.

Corollary 4.14 (Taylor [33]). If every normal and fine ideal on κ = µ+ is (µ, κ)-regular,

then every κ-complete ideal on κ is (µ, κ)-regular.

Proof. Suppose I is a κ-complete ideal on κ, and let A ∈ I+. If I � A is nowhere κ+-

saturated, then I � A is (2, κ)-regular by Lemma 4.11. Otherwise, Lemma 4.12 implies there

is some B ⊆ A such that I � B is isomorphic to a normal ideal N . By hypothesis, N is

(µ, κ)-regular, and this is clearly preserved by isomorphisms. Thus by Lemma 4.13, I is

(µ, κ)-regular.

Corollary 4.15. The normality assumption can be replaced by κ-completeness in Proposi-

tion 3.6.

Proof. Assume κ = µ+, µ<cf(µ) = µ, and I is a κ-complete, κ+-saturated ideal on κ. Suppose

for a contradiction that there is A ∈ I+ such that [A] “β < cf(µ), f : β → Ord, and

f /∈ V .” By lemma 4.12, there is a B ⊆ A such that I � B is isomorphic to a normal ideal N

on κ. By Proposition 3.6, forcing with P(κ)/N adds no new functions with domain β and

range in the ordinals, so the same is true of I � B.

55

Lemma 4.16. Let µ < κ be regular cardinals, and λ ≥ κ such that λ<µ = λ. If I is

a normal, fine, κ-complete, nowhere κ+-complete, λ+-saturated ideal on Z ⊆ P(λ), and

P(Z)/I is µ-distributive, then I is regular iff I is (µ, λ)-regular.

Proof. Fix a bijection f : λ → [λ]<µ. Let j : V → M be a generic embedding arising

from forcing with P(Z)/I. If x ⊆ j[λ] has cardinality < µ in M , then by µ-distributivity,

j−1[x] ∈ V and j(j−1[x]) = x. j−1[x] = f(α) for some α < λ, so x = j(f)(j(α)). Also,

for any α < λ, j(f)(j(α)) = j(f(α)) ⊆ j[λ]. Thus, by Los’s theorem, A = {z : f � z is a

bijection between z and [z]<µ} ∈ I∗.

If {Aα : α < λ} ⊆ I+, let {Bα : α < λ} ⊆ I+ be a refinement such that for any X ⊆ λ

of order type µ,⋂α∈X Bα = ∅. We may assume each Bα ⊆ α ∩ A. For each z ∈ Z, let

s(z) = {α : z ∈ Bα}. For all z, s(z) ∈ [z]<µ. For each Bα, there is a constant ηα such that

Cα = {z ∈ Bα : s(z) = f(ηα)} ∈ I+.

Suppose z ∈ Cα0 ∩ Cα1 . Then {α0, α1} ⊆ s(z) = f(ηα0) = f(ηα1). If z′ ∈ Cα0 , then

s(z′) = f(ηα0) = f(ηα1), so z′ ∈ Cα1 . Thus Cα0 ⊆ Cα1 . Switching the roles of α0 and α1, we

conclude Cα0 = Cα1 . Therefore for all α1 < α2 < λ, either Cα0 ∩ Cα1 = ∅ or Cα0 = Cα1 .

For C ⊆ Z, Let t(C) = {α : C = Cα}. If α ∈ t(C) and z ∈ C, then t(C) ⊆ s(z), so

|t(C)| < µ. For each C ∈ {Cα : α < κ}, choose a splitting of C into disjoint I-positive sets

{DCξ : ξ < ot(t(C))}. (This is possible by the assumptions: the (µ, λ)-regularity of I implies

that I is nowhere-prime. If I∗∩P(Y ) were an ultrafilter on Y ⊆ Z, then by κ-completeness,

it would not be (α, α)-regular for any α < κ. µ-distributivity implies that 2<µ < κ, and so

I is nowhere µ-saturated.) By induction, define Eα ⊆ Cα to be DCξ , where C = Cα, and ξ is

the least ordinal such that DCξ 6= Eγ for all γ < α. Then {Eα : α < κ} is a pairwise disjoint

refinement of {Aα : α < κ}.

Theorem 4.17. Suppose κ = µ+, µ<cf(µ) = µ, and I is a κ-complete ideal on κ. Then I is

(cf(µ), κ)-regular iff it is regular.

56

Proof. Suppose I is a κ-complete, (cf(µ), κ)-regular ideal on κ = µ+. Let A ∈ I+. If I � A

is nowhere κ+-saturated, then I � A is regular by Lemma 4.11. Otherwise, Lemma 4.12

implies there is some B ⊆ A and some bijection f on κ such that f∗(I � A) = N is a

κ+-saturated normal ideal on κ. N is (cf(µ), κ)-regular, and by Proposition 3.6, P(κ)/N is

cf(µ)-distributive. Hence, N is regular by Lemma 4.16. Thus, {[C] : I � C is regular} is

dense, so Lemma 4.13 implies that I is regular. The other direction is trivial.

To show the necessity of assuming the ideals are normal in clause (2) of Taylor’s theorem,

we use the following result of Shelah:

Theorem 4.18 (Shelah [31]). Suppose there is a cardinal κ such that {α < κ : α is

supercompact} is stationary in κ. Then there is a forcing extension in which:

1. P(ω1)/NS ∼= B(Col(ω,< ω2)).

2. The algebra P(ω1)/NS is the union of ω1 many countably complete filters.

Proposition 4.19. In Shelah’s model, there is no ω1-dense, countably complete ideal on ω1,

but there is a set of countably complete ideals {Iα : α < ω1} such that P(ω1) =⋃

(Iα ∪ I∗α).

Proof. Let P(ω1)/NS =⋃{Fα : α < ω1}, where each Fα is a countably complete filter. For

each α, let Iα = {X : [ω1 \X] ∈ Fα}.

If there were a countably complete, ω1-dense ideal on ω1 in Shelah’s model, then there would

be a normal one J . In the model, NSω1 is ω2-saturated, so by Lemma 4.3, J = NSω1 � A

for some stationary set A. But Col(ω,< ω2) is nowhere ω1-dense.

Proposition 4.20. Suppose NSω1 is ω2-saturated but nowhere ω1-dense. Let J be a of ω1

many countably complete ideals on ω1 such that every subset of ω1 is measurable by one of

them. J must have ω1 many non-normal, nowhere ω2-saturated members.

57

Proof. Let J0 = {I ∈ J : Iα is nowhere ω2-saturated}. Since NSω1 is ω2-saturated,

Lemma 4.3 implies that every member of J0 is not normal. Let {Iα : α < ω1} enumer-

ate J \ J0, with repetitions in case this set is countable. For each Iα, use Lemma 4.12 to

pick (Aα, fα, Nα) such that Aα ∈ I+α , fα is a bijection on ω1, Nα is a normal ω2-saturated

ideal, and Nα = (fα)∗(Iα � Aα). Let J1 = {Iα � Aα : α < ω1}, and let J =⋂J1.

For each α, let Jα = (fα)∗(J). So Jα ⊆ Nα, and both are ω2-saturated. Let {[Bβ] : β < ω1}

be a maximal antichain in P(ω1)/Jα where each Bβ ∈ Nα. Let Cα = ∇β<ω1Bβ ∈ Nα. If

X ∈ J+α and X ⊆ Cα, then X ∈ Nα, so by maximality X ∩Bβ ∈ J+

α for some β. Thus, Cα is

≤Jα-maximal among all sets in Nα, so Nα = Jα � (ω1 \ Cα). Since fα is a bijection, we have

for each α a set Dα such that Iα � Aα = J � Dα.

J is nowhere ω1-dense, so by Theorem 4.1, J is regular. Let {Eα : α < ω1} be a disjoint

refinement of {Dα : α < ω1} into J-positive sets. Split each Eα into disjoint J-positive sets

{Eβα : β < ω1}, and let Fα =

⋃β<ω1

Eβα. Then the collection {Fα : α < ω1} is pairwise

disjoint, and every member is (J � Dγ)-positive for all relevant γ.

Assume now for a contradiction that J0 is countable. For each even ordinal α, let Gα =

Fα ∪ Fα+1. At most one Gα can be in I∗ for any I ∈ J0, so let β be such that ω1 \ Gβ is

I-positive for all I ∈ J0. By the Alaoglu-Erdos theorem, there are disjoint X0, X1 ⊆ ω1 \Gβ,

both I-positive for all I ∈ J0. Then X0 ∪ Fβ and X1 ∪ Fβ+1 are disjoint sets which are

positive for all I ∈ J . This contradicts the assumption that every subset of ω1 is measurable

by some I ∈ J .

58

Chapter 5

Consistency results from generic large

cardinals

Our main goal in this chapter is to pull apart density and nonregularity at higher cardinals,

showing that Theorem 4.1 and Corollary 4.6 are indeed specific to ω1. Rather than creating

a new construction from large cardinals, we start from generic large cardinal assumptions

shown consistent in Chapter 2. A potential advantage of this approach is that if the con-

sistency strength of dense ideals on spaces other than ω1 is ever reduced below almost-huge

cardinals, the following arguments will be applicable to those contexts as well. As a conse-

quence of the techniques, we derive some new results concerning the mutual inconsistency

of some generic large cardinals and strengthen a result of Woodin. We also show that the

kind of generic supercompactness shown consistent in Chapter 2 is also consistent with �

holding very often.

59

5.1 Foreman’s Duality Theorem

Our main tool will be Foreman’s Duality Theorem [14], which allows a precise characteriza-

tion of the effect of forcing on the structure of precipitous ideals. We present here a slight

generalization of Foreman’s theorem, correcting a minor mistake in [14]. The mistake was

not in the main result but in a statement regarding the extent of the applicability of its

hypothesis. The correction leads to a more general theorem and an equivalence between the

generic extendibility of elementary embeddings and a structural characterization of induced

ideals.

Claim (Foreman). Suppose I is a precipitous ideal on Z, P is a partial order, and m is a

P(Z)/I-name such that (1, m) P(Z)/I∗j(P) “j−1[H] is P-generic over V ,” where H denotes

the generic for j(P). Then there is some q ∈ P such that the map e defined by p 7→ (1, m ∧˙j(p)) is a regular embedding of P � q into P(Z)/I ∗ (j(P � q) � m).

Counterexample Assume CH and there is an ω1-dense ideal I on ω1. Then P(ω1)/I ∼=

B(Col(ω, ω1)). Since Col(ω, ω1) ∼ Add(ω1) × Col(ω, ω1) under CH, forcing with P(ω1)/I

adds a Cohen-generic subset H ⊆ ω1. Let G be generic for P(ω1)/I and let j : V →

M ⊆ V [G] be the generic ultrapower embedding. Then H ∈ M , and H is a condition

m ∈ j(Add(ω1)) = Add(ω1)V [G]. If we take a P(ω1)/I-name m for m, then the condition

(1, m) ∈ P(ω1)/I ∗ ˙Add(ω1) forces that j−1[H] is Add(ω1)-generic over V , where H denotes

the generic for Add(ω1)V [G].

Now the map e defined by e(p) = (1, ˙j(p) ∧ m) = (1, p ∧ m) is not a regular embedding

of Add(ω1) into P(ω1)/I ∗ Add(ω1) for one simple reason. Its range is not contained in

the purported codomain. For any nontrivial p ∈ Add(ω1), [ω1]I does not force that p is

compatible with m. In fact, the set {(p,A) : A p ⊥ m} is dense in the product order of

Add(ω1)× P(ω1)/I. Thus Foreman’s claim is incorrect.

60

To see this, let ι : Add(ω1) × Col(ω, ω1) → P(ω1)/I be a dense embedding, and let the

name for the condition m be the projection of ι−1[G] to the first coordinate. Let (p,A) ∈

Add(ω1)×P(ω1)/I be arbitrary, and let ι(p0, q0) ≤ [A]I . If p0 ⊥ p, then any [B]I ≤ ι(p0, q0)

forces m ⊥ p. Otherwise, let p1, p2 ≤ p0 ∧ p be such that p1 ⊥ p2. Let B ⊆ A be such that

[B]I ≤ ι(p1, q0). Then B p2 ⊥ m.

To fix this, we only need to redefine the map e. We should instead send p to ||j(p) ∈ H||.

In some cases, this will coincide with Foreman’s map, but not always.

Theorem 5.1. Suppose I is a precipitous ideal on Z and P is a boolean algebra. Let j :

V → M ⊆ V [G] denote a generic ultrapower embedding arising from I. Suppose Q is a

P(Z)/I-name for a forcing and H0 is a name such that:

(1) 1 P(Z)/I∗Q H0 is j(P)-generic over M ,

(2) 1 P(Z)/I∗Q j−1[H0] is P-generic over V , and

(3) for all p ∈ P, 1 1P(Z)/I∗Q j(p) /∈ H0.

In V [G], let K be the ideal {p ∈ j(P) : 1 Q p /∈ H0}. There is P-name for an ideal J on Z

and a canonical isomorphism

ι : B(P ∗ ˙P(Z)/J) ∼= B(P(Z)/I ∗ ˙j(P)/K).

Proof. Denote a generic for P(Z)/I ∗ j(P)/K by G ∗ h. In V [G ∗ h], let H = {p ∈ j(P) :

[p]K ∈ h}. First we claim H has properties (1),(2),(3) assumed for H0.

(1) If D ∈M is open and dense in j(P), then {[d]K : d ∈ D and d /∈ K} is dense in j(P)/K.

For otherwise, there is p ∈ j(P)\K such that p∧d ∈ K for all d ∈ D. By the definition of

K, we can force with Q to obtain a filter H0 ⊆ j(P) with p ∈ H0. But H0 cannot contain

61

any elements of D, so it is not generic over M , a contradiction. Thus if h ⊆ j(P)/K is

generic over V [G], then H is j(P)-generic over M .

(2) If A ∈ V is a maximal antichain in P, then {[j(a)]K : a ∈ A and j(a) /∈ K} is a maximal

antichain in j(P)/K. For otherwise, there is p ∈ j(P) \K such that p ∧ j(a) ∈ K for all

a ∈ A. We can force with Q to obtain a filter H0 ⊆ j(P) with p ∈ H0. But H0 cannot

contain any elements of j[A], so j−1[H0] is not generic over V , a contradiction.

(3) If p ∈ P, then the assumption about H0 implies that 1 1P(Z)/I j(p) ∈ K.

Let e : P → B(P(Z)/I ∗ j(P)/K) be defined by p 7→ ||j(p) ∈ H||. By (3), this map has

trivial kernel. By elementarity, it is an order and antichain preserving map. If A ⊆ P is a

maximal antichain, then it is forced that j−1[H] ∩ A 6= ∅. Thus e is regular.

Whenever H ⊆ P is generic, there is a further forcing yielding a generic G ∗ h ⊆ P(Z)/I ∗

j(P)/K such that j[H] ⊆ H. Thus there is an embedding j : V [H] → M [H] extending

j. In V [H], let J = {A ⊆ Z : 1 (P(Z)/I∗j(P)/K)/e[H] [id]M /∈ j(A)}. In V , define a map

ι : P∗ ˙P(Z)/J → B(P(Z)/I ∗ ˙j(P)/K) by (p, A) 7→ e(p)∧||[id]M ∈ j(A)||. It is easy to check

that ι is order and antichain preserving.

We want to show the range of ι is dense. Let (B, q) ∈ P(Z)/I ∗ ˙j(P)/K, and WLOG, we

may assume there is some f : Z → V in V such that B q = [[f ]M ]K . By the regularity

of e, let p ∈ P be such that for all p′ ≤ p, e(p′) ∧ (B, q) 6= 0. Let A be a P-name such that

p A = {z ∈ B : f(z) ∈ H}, and ¬p A = Z. 1 P A ∈ J+ because for any p′ ≤ p, we

can take a generic G ∗ h such that e(p′) ∧ (B, q) ∈ G ∗ h. Here we have [id]M ∈ j(B) and

[f ]M ∈ H, so [id]M ∈ j(A). Furthermore, ι(p, A) forces B ∈ G and q ∈ h, showing ι is a

dense embedding.

Proposition 5.2 (Foreman). Suppose the ideal K in Theorem 5.1 is forced to be principal.

Let m be such that P(Z)/I K = {p ∈ j(P) : p ≤ ¬m}. Suppose f and A are such that

62

A m = [f ], and B is a P-name for {z ∈ A : f(z) ∈ H}. Let I be the ideal generated by I

in V [H]. Then I � B = J � B, and A \B ∈ J .

Proof. Clearly J ⊇ I. Suppose that C ⊆ B and C ∈ I+, and let p ∈ P be arbitrary.

WLOG P is a complete boolean algebra. For each z ∈ Z, let bz = ||z ∈ C||. In V , define

C ′ = {z : p ∧ bz ∧ f(z) 6= 0}. p C ⊆ C ′, so C ′ ∈ I+. If G ⊆ P(Z)/I is generic with

C ′ ∈ G, then j(p)∧ b[id]∧ m 6= 0. Take H ⊆ j(P) generic over V [G] with j(p)∧ b[id]∧ m ∈ H.

Since b[id] Mj(P) [id] ∈ j(C), p 1 C ∈ J as p ∈ H = j−1[H]. Thus J � B = I � B.

Furthermore, if H ⊆ P is any generic, and G ∗ h is generic extending e[H] with A ∈ G, then

m = j(f)([id]) ∈ H, so [id] ∈ j(B). Thus A \B ∈ J .

Corollary 5.3. Let I be a precipitous ideal on Z and P a partial order. Let I denote the

ideal generated by I in V P. The following are equivalent:

(1) In some P(Z)/I ∗ j(P)-generic extension V [G ∗ H], H = j−1[H] is P-generic over V .

(2) For some conditions a, b, there is an isomorphism ι : B(P ∗ P(Z)/I) � a ∼= B(P(Z)/I ∗

j(P)) � b such that ι(||p ∈ H||) = ||j(p) ∈ H|| for all p ∈ P.

Proof. WLOG we may assume P is a complete boolean algebra. (1) implies that for some

(A, m) ∈ P(Z)/I ∗ j(P), (A, m) j−1[H] is P-generic over V . By shrinking A if necessary,

we may assume for some f ∈ V , A m = [f ]. Let e : P → B(P(A)/I ∗ j(P) � m) be

defined by p 7→ ||j(p) ∈ H||. We claim that for some p ∈ P, e(p′) 6= 0 for all p′ ≤ p.

Otherwise, the set of p such that e(p) = 0 is dense, so whenever G ∗ H is generic with

(A, m) ∈ G ∗ H, there is p ∈ P such that j(p) ∈ H and ||j(p) ∈ H|| = 0, a contradiction. If

D = {p ∈ P : (∀p′ ≤ p)e(p′) 6= 0}, then p0 =∑D ∈ D. Since {p ≤ ¬p0 : e(p) = 0} is dense

below ¬p0, we must have e(¬p0) = 0. Thus (A, m) j(p0) ∈ H, and so A m ≤ ˙j(p0).

The hypotheses of Theorem 5.1 are satisfied with respect to I � A and P � p0, and the ideal K

on j(P � p0) is the principal ideal generated by ¬m. We get an isomorphism ι : B((P � p0) ∗

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˙P(A)/J) ∼= P(A)/I ∗ ˙(j(P � p0) � m)) such that ι(||p ∈ H||) = ι(p, 1) = e(p) = ||j(p) ∈ H||

for all p ∈ P � p0. By Proposition 5.2, there is a set B ∈ V P such that we may replace

P(A)/J on the left by P(B)/I. Thus (1) ⇒ (2). For the other direction, if an isomorphism

exists with those properties, then for any generic G ∗ H, ι−1[G ∗ H] ∩ P = j−1[H].

Corollary 5.4 (Foreman). If I is a κ-complete precipitous ideal on Z and P is κ-c.c., then

there is a canonical isomorphism ι : P ∗ P(Z)/I ∼= P(Z)/I ∗ j(P).

Proof. If G ∗ H ⊆ P(Z)/I ∗ j(P) is generic, then for any maximal antichain A ⊆ P in V ,

j[A] = j(A), and M |= j(A) is a maximal antichain in j(P). Thus j−1[H] is P-generic over

V , and clearly for each p ∈ P, we can take H with j(p) ∈ H. Thus Theorem 5.1 implies that

for some J , K, we have an isomorphism ι : B(P ∗ P(Z)/J) → B(P(Z)/I ∗ j(P)/K). In this

case, K is trivial, and so Proposition 5.2 implies that J = I.

Proposition 5.5. If Z, I,P, J,K, ι are as in Theorem 5.1, then whenever H ⊆ P is generic,

J is precipitous and has the same completeness and normality that I has in V . Also, if

G ⊆ P(Z)/J is generic and G ∗ h = ι[H ∗ G], then if j : V [H] → M [H] is as above,

M [H] = V [H]Z/G and j is the canonical ultrapower embedding.

Proof. Suppose H∗G ⊆ P∗P(Z)/I is generic, and let G∗h = ι[H∗G] and H = {p : [p]K ∈ h}.

For A ∈ J+, A ∈ G iff [id]M ∈ j(A). If i : V [H] → N = V [H]Z/G is the canonical

ultrapower embedding, then there is an elementary embedding k : N → M [H] given by

k([f ]N) = j(f)([id]M), and j = k ◦ i. Thus N is well-founded, so J is precipitous. If

f : Z → Ord is a function in V , then k([f ]N) = j(f)([id]M) = [f ]M . Thus k is surjective

on ordinals, so it must be the identity, and N = M [H]. Since i = j and j extends j, i and

j have the same critical point, so the completeness of J is the same as that of I. Finally,

since [id]N = [id]M , I is normal in V iff J is normal in V [H], because j �⋃Z = j �

⋃Z,

and normality is equivalent to [id] = j[⋃Z].

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Proposition 5.6. Suppose ι : B(P ∗P(Z)/J)→ B(P(Z)/I ∗ j(P)/K) is as in Theorem 5.1.

For any B ∈ I+, if (p, A) ∈ P ∗ P(Z)/J and p A = B, then ι(p, A) = (B, 1) ∧ e(p).

Proof. By definition, ι(p, A) = e(p) ∧ ||[id] ∈ j(A)||. If G ∗ h is generic with this condition,

then [id] ∈ j(Ae−1[G∗h]), and so [id] ∈ j(B) and (B, 1)∧ e(p) ∈ G ∗h. If (B, 1)∧ e(p) ∈ G ∗h,

then [id] ∈ j(B) = j(Ae−1[G∗h]), so ι(p, A) ∈ G ∗ h. The equality follows.

5.2 Preservation and destruction

We show here that density and nonregularity can be consistently separated by showing that

certain forcings will preserve nonregularity while destroying density. We give two different

preservation results, each of which starts from different assumptions about the ground model.

They provide two different paths to our desired consistency result.

Lemma 5.7. Suppose µ is regular, P is µ-c.c., I is an ideal on Z, and p P “I is (µ, κ)-

regular.” Then I is (µ, κ)-regular.

Proof. Let 〈Aα〉α<κ ⊆ I+. Let 〈Bα〉α<κ be such that p P “〈Bα〉α<κ ⊆ I+ is a refinement

of 〈Aα〉α<κ such that for all X ⊆ κ of size µ,⋂α∈X Bα = ∅.” For each α, let Cα = {z :

(∃q ≤ p)q z ∈ Bα}. Each Cα ∈ I+ because p Bα ⊆ Cα. For each z ∈ Z, let

s(z) = {α : z ∈ Cα}. Since P is µ-.c.c., |s(z)| < µ for each z. So 〈Cα〉α<κ is the desired

refinement of 〈Aα〉α<κ.

Lemma 5.8. Suppose I is a normal, fine, nonregular ideal on Z ⊆ P(λ) that is λ+-saturated

but nowhere λ-saturated. Let 〈Aα〉α<λ ⊆ I+ be a sequence with no disjoint refinement into

I-positive sets. Then there is an I-positive B ∈ I+ such that for all I-positive C ⊆ B,

|{α : C ∩ Aα ∈ I+}| = λ.

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Proof. Suppose towards a contradiction that {B : |{α : B ∩ Aα ∈ I+}| < λ} is dense. Let

{Bα : α < λ} be a maximal antichain contained in this collection. We may assume the Bα’s

are pairwise disjoint. Let s(α) = {β : Bα ∩ Aβ ∈ I+}. By Theorem 1.5, for each Bα, there

is a disjoint refinement of {Bα ∩ Aβ : β ∈ s(α)} into I-positive sets {Cαβ : β ∈ s(α)}, as

{I � (Bα ∩ Aβ) : β ∈ s(α)} is a set of < λ many nowhere λ-saturated ideals on Bα. Since

{Bα : α < λ} is a maximal antichain, for all β there is α such that β ∈ s(α). Let h(β) = the

least α such that β ∈ s(α). Then {Ch(β)β : β < λ} is a disjoint refinement of {Aβ : β < λ}

into I-positive sets, contrary to assumption.

Lemma 5.9. Suppose µ < λ are regular cardinals, κ = µ+, P is µ-c.c., and I is a normal,

fine, nonregular, κ-complete ideal on Z ⊆ P(λ). Then P I is nonregular.

Proof. Suppose 〈Aα〉α<λ witnesses that I is nonregular. Let X0 = {α < λ : I � Aα is nowhere

λ+-saturated}. For each α ∈ X1 = λ \X0, pick A′α ⊆ Aα such that I � A′α is λ+-saturated.

Note that I � (⋃α∈X1

A′α ∩ α) is λ+-saturated. We claim that {A′α : α ∈ X1} witnesses that

I is nonregular.

To see this, assume towards a contradiction that {A′′α : α ∈ X1} is a pairwise disjoint

refinement into I-positive sets. By Lemma 4.2, there is a collection {Bα : α < λ+} such

that (∀α < λ+)(∀β ∈ X0)Bα ∩ Aβ ∈ I+, and (∀α < β < λ+)Bα ∩ Bβ is nonstationary. Let

γ < λ+ be such that Cα = A′′α \ Bγ ∈ I+ for all α ∈ X1. Apply Lemma 4.2 again to split

Bγ into pairwise disjoint subsets {Cα : α ∈ X0} such that Cα ∈ I+ and Cα ⊆ Aα for all

α ∈ X0. Then {Cα : α < λ} is a pairwise disjoint refinement of {Aα : α < λ}, contrary to

assumption.

Considering I � (⋃α∈X1

A′α ∩ α) and renaming the A′α’s, we can assume WLOG that we

start with a λ+-saturated, normal, fine, κ-complete, nonregular ideal I on Z with 〈Aα〉α<λ

witnessing the nonregularity. Assume for the sake of a contradiction that p I is regular.

Let 〈Bα〉α<λ be a sequence of P-names such that p 〈Bα〉α<λ is a pairwise disjoint refinement

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of 〈Aα〉α<λ into I-positive sets. So {(p, Bα) : α < λ} is an antichain in P ∗ P(Z)/I.

Let ι : P ∗ P(Z)/I → B(P(Z)/I ∗ j(P) be given by Corollary 5.4. For each α < λ, choose

(Cα, qα) ≤ ι(p, Bα). Since

(Cα, qα) ≤ ι(p, Bα) ≤ ι(p, Aα) ≤ ι(1, Aα) = (Aα, 1),

we have Cα ⊆I Aα for each α < λ. Since 〈Aα〉α<λ has no disjoint refinement into I-positive

sets, neither does 〈Cα〉α<λ.

Since I satisfies the hypotheses of Lemma 5.8, there is D ∈ J+ such that for all J-positive

E ⊆ D, |{α : E ∩Cα ∈ J+}| = λ. Thus, if we take a generic G ⊆ P(Z)/I with D ∈ G, then

a density argument shows that {α : Cα ∈ G} is unbounded in λ. Let M = V Z/G. M thinks

that µ is a regular cardinal and j(P) is µ-c.c. By the closure properties of M , these hold in

V [G] as well. By Theorem 3.4, cf(λ) ≥ µ in V [G], so there are at least µ many Cα’s in G.

Since (Cα, qα) ⊥ (Cβ, qβ) for α 6= β, we must have that qGα ⊥ qGβ for distinct Cα, Cβ ∈ G.

However, these are all elements of a µ-c.c. partial order, so we have a contradiction. Thus

P I is nonregular.

Definition. If P is a partial order and Z ⊆ P(X), we will say P is Z-absolutely κ-c.c. when

for all normal, fine, |X|+ saturated ideals I on Z, P(Z)/I j(P) is j(κ)-c.c.

Lemma 5.10. Suppose κ = µ+ and Z ⊆ {z ∈ Pκ(λ) : z ∩ κ ∈ κ} is stationary. Suppose P

is Z-absolutely κ-c.c. and κ ≤ d(P � p) ≤ λ for all p ∈ P. Then in V P, there are no normal,

fine, κ-complete λ-dense ideals on Z.

Proof. Suppose p J is a normal, fine, κ-complete, λ+-saturated ideal on Z. Let I =

{X ⊆ Z : p X ∈ J}. It is easy to check that I is normal and fine. The map σ :

P(Z)/I → B(P � p ∗ P(Z)/J) that sends X to (||X ∈ J+||, ˙[X]J) is an order-preserving and

antichain-preserving map, so I is λ+-saturated.

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Let H be P-generic over V with p ∈ H. Since P is κ-c.c., I remains normal. By Corollary 5.4,

PV [H](Z)/I ∼= (PV (Z)/I ∗ j(P))/e[H]. Thus I is normal, fine, and λ+-saturated, and I ⊆

J . By Lemma 4.3, there is A ∈ I+ such that J = I � A. Since j(κ) = λ+, j(P) is

forced to be nowhere λ-dense, and thus P(Z)/I ∗ j(P) is nowhere λ-dense. Since d(P) ≤ λ,

(PV (Z)/I ∗ j(P))/e[H] is nowhere λ-dense. Thus J is not λ-dense.

Theorem 5.11. If almost-huge cardinals are consistent, then ZFC does not prove the analogs

of Theorem 4.1 and Corollary 4.6 with ω1 is replaced by µ+ where µ > ω is regular. If

ω < µ < κ ≤ λ < δ are regular, and κ carries an almost-huge tower of height δ, then there is

a forcing extension in which κ = µ+, there are no normal, fine, κ-complete, λ-dense ideals

on a stationary Z ⊆ Pκ(λ), but there is a normal, fine, κ-complete, nonregular ideal on Z.

If κ is super-almost-huge, then there is a forcing extension in which κ = µ+, and for every

regular λ ≥ κ, there is a stationary Zλ ⊆ Pκ(λ) such that there are no dense ideals on Zλ,

but there is a nonregular ideal on Zλ.

Proof. Let µ < κ ≤ λ be as hypothesized. By Theorem 2.17, there is a forcing extension in

which κ = µ+ there is a normal, fine, κ-complete, λ-dense ideal I on Pκ(λ). In this model,

let P be any µ-c.c. partial order such that κ ≤ d(P � p) ≤ λ for all p ∈ P, such as Add(ω, κ).

If J is any normal, fine, κ-complete, λ+-saturated ideal on Pκ(λ), and j : V → M ⊆ V [G]

is a generic embedding arising from J , then M |= j(P) is µ-c.c. Since Mµ ∩ V [G] ⊆M , j(P)

is µ-c.c. in V [G]. Hence by Lemma 5.10, we destroy all dense ideals by forcing with P.

By Lemma 5.9, the generated ideal I remains nonregular. Furthermore, one can check

that the ideal given by Theorem 2.17 is µ-distributive, so it is actually (µ, λ)-nonregular by

Lemma 4.16. So we may alternatively apply Lemma 5.7 to see that I is (µ, λ)-nonregular

and thus nonregular.

If we start with a super-almost-huge κ, then in a forcing extension κ = µ+ and these ideals

on Pκ(λ) exist for all regular λ ≥ κ, so forcing with P has the these effects simultaneously

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with respect to all regular λ ≥ κ.

At present, it is unknown whether properties (1) and (2) from Theorem 4.1 are equivalent

under ZFC when ω1 is replaced by larger cardinal. Does the existence of κ many normal and

fine ideals on κ that collectively measure all subsets of κ imply the existence of a κ-complete,

κ-dense ideal on κ? The best we can do now is point to the need for a different approach–our

methods are demonstrably inadequate:

Proposition 5.12. Suppose GCH, µ < κ = µ+ ≤ λ, there is a normal, fine, κ-complete,

λ-dense ideal I on Z ⊆ Pκ(λ), and P is Z-absolutely κ-c.c. Then in V P, either there is a

normal, fine, κ-complete, λ-dense ideal on Z, or for every set {Iα : α < λ} of normal, fine,

κ-complete ideals on Z, there is A ⊆ Z that is nonmeasurable for each Iα.

Proof. If there is p ∈ P such that d(P � p) < κ, then p I is λ-dense. Otherwise, d(P � p) ≥ κ

for all p ∈ P. Suppose p {Iα : α < λ} is a set of normal, fine, κ-complete ideals on Z such

that every A ⊆ Z is measurable in one of them. By Lemma 4.7, there is some name I such

that p I is a normal, fine, κ-complete, λ+-saturated ideal on Z such that ˙P(Z)/I has a

weakly dense subset {Aα : α < λ}. In V , let J = {A ⊆ Z : p A ∈ I}. J is normal, fine,

κ-complete, and λ+-saturated. By the hypothesis on P, Corollary 5.4, and Lemma 4.3, there

is some name B such that p I = J � B.

Let θ > λ be regular such that P ∈ Hθ. Let M ≺ Hθ be such that {P, p, I, B, {Aα : α <

λ}} ∈M , λ ⊆M , |M | = λ, and M<κ ⊆M . If A ⊆ P∩M is an antichain, then by the κ-c.c.

of P, A ∈M . If A is not maximal in P, then by elementarity, there is some p ∈M such that

A ⊥ p, so A is not maximal in P∩M . Thus P0 = P∩M is a regular suborder of P. If G ⊆ P

is generic, then BG and each AGα are in V [G ∩ P0]. Let J0 be the ideal generated by J in

V [G ∩ P0]. If C ∈ J0 � B, then {Aα : α < λ} is weakly dense in P(Z)/J0. This is because if

C ∈ P(Z)V [G∩P0], then in V [G], there is some Aα such that either Aα ≤J C or Aα ≤J Z \C,

and this property is absolute between transitive models.

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However, by Lemma 5.10, forcing with P0 destroys all dense ideals on Z. By the κ-c.c., and

GCH in V , we have P0 2<λ = λ. Thus by Theorem 4.8, there cannot exist a normal, fine,

λ+-saturated ideal K on Z in V [G ∩ P0] such that P(Z)/K has a weakly dense subset of

size λ. This contradiction establishes the dichotomy.

In contrast to density and saturation, nonregularity of ideals can be destroyed by small

forcing:

Proposition 5.13. Suppose κ = µ+ and Z ⊆ Pκ(λ) is such that there is a normal, fine,

κ-complete, nonregular ideal on Z, but there are no normal, fine, κ-complete, λ-dense ideals

on Z. Then forcing with Col(ω, µ) destroys all nonregular ideals on Z.

Proof. Assume towards a contradiction that G ⊆ Col(ω, µ) is generic, and J ∈ V [G] is

a normal, fine, κ-complete, nonregular ideal on Z. By the first part of the argument for

Lemma 5.9, we may assume J is λ+-saturated. Let p force these properties for J , and in V ,

let I = {A ⊆ Z : p A ∈ J}. I is normal, fine, κ-complete, and λ+ saturated, and properties

are preserved for I in V [G], so by Lemma 4.3, J = I � A for some A. By Lemma 4.5, there

must be some J-positive B ⊆ A such that J � B = I � B is λ-dense. Let p 〈Bα〉α<λ is

dense in I � B.

By κ-completeness, there is some q ≤ p, q ∈ G, such that C = {z : q z ∈ B} ∈ I+. For

each r ≤ q and α < λ let Crα = {z ∈ C : r z ∈ Bα}. In V , let D ⊆ C be I-positive.

In V [G], there is some Bα ⊆J D, and by κ-completeness, there is some r ∈ G such that

Crα ∈ J+. Thus Cr

α ∈ I+ and Crα \D ∈ I. We have that {Cr

α : r ≤ q, α < λ, and Crα ∈ I+}

is dense below C, contrary to assumption.

Regardless of the existence of dense ideals, we can destroy nonregular ideals while preserving

saturation:

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Proposition 5.14. Suppose λ ≥ κ = µ+, µ<µ = µ, and Z ⊆ Pκ(λ) is stationary. Forcing

with Add(µ, κ) destroys all κ-complete, normal, fine, nonregular ideals on Z, while preserving

the existence of κ-complete, normal, fine, λ+-saturated ideals on Z.

Proof. First note that µ<µ = µ implies Add(µ, κ) is κ-c.c. If j : V → M ⊆ V [G] is a

generic embedding arising from a κ-complete, normal, fine, λ+-saturated ideal on Z, then

j(Add(µ, κ)) = Add(µ, j(κ))M = Add(µ, j(κ))V [G], which is j(κ)-c.c. in V [G], and j(κ) =

(λ+)V . Thus by Corollary 5.4, and Proposition 5.5 if I is a κ-complete, normal, fine, λ+-

saturated ideal on Z, then I has these properties in V Add(µ,δ).

On the other hand, suppose p0 Add(µ,κ) J is κ-complete, normal, fine, nonregular ideal on Z.

By the first part of the proof for Lemma 5.9, we may assume p also forces J is λ+-saturated.

Let I = {A ⊆ Z : p0 A ∈ J}. Then I is normal, fine, κ-complete, and λ+-saturated.

Whenever H ⊆ Add(µ, δ) is generic, JH = I � A for some A.

Let e : Add(µ, κ) → B(P(Z)/I ∗ Add(µ, j(κ))) be the embedding from Theorem 5.1. If

H ⊆ Add(µ, κ) is generic, then there is an isomorphism σ : B(P(Z)/I ∗Add(µ, j(κ))/e[H]→

PV [H](Z)/I. Suppose 〈Aα〉α<λ is a sequence of J-positive sets. For each Aα, choose (Bα, qα) ∈

PV (Z)/I ∗Add(µ, j(κ)) such that σ(Bα, qα) ≤ Aα. By the λ+-c.c., there is some β < λ+ such

that PV (Z)/I

⋃α<λ dom(qα) ⊆ β. Since PV (Z)/I |λ| = µ, there is name for an antichain

〈rα〉α<λ in Add(µ, j(κ)) such that dom(rα)∩β = ∅ for all α < λ. Then 〈σ(Bα, qα∧ rα)〉α<λ

is a disjoint refinement of 〈Aα〉α<λ into J-positive sets.

5.3 Compatibility with square

Solovay [32] showed that �δ fails when δ ≥ κ and κ is supercompact. In contrast, we’ll show

(∀δ ≥ κ)�δ is consistent with the kind generically supercompact κ constructed in Chapter

2. Ironically, a common feature between traditional and generic supercompactness can show

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the failure of � in the traditional case, and allow � to be forced in the generic case while

preserving generic supercompactness. The key difference is that nontrivial forcings may be

absorbed into the quotient algebras of the ideals in the generic case.

For a cardinal δ, let Sδ be the collection of bounded approximations to a �δ sequence. That

is, a condition is a sequence 〈Cα : α ∈ η ∩ Lim〉 such that η < δ+ is a successor ordinal,

each Cα is a club subset of α of order type ≤ δ, and whenever β is a limit point of Cα,

Cα ∩ β = Cβ. For proof of the following lemma, we refer the reader to [8].

Lemma 5.15. For every cardinal δ, Sδ is countably closed and (δ + 1)-strategically closed

and adds a �δ sequence 〈Cα : α ∈ δ+ ∩ Lim〉 =⋃G, where G ⊆ Sδ is the generic filter. For

every regular λ ≤ δ, there is a Sδ-name for a “threading” partial order Tλδ that adds a club

C ⊆ (δ+)V of order type λ and such that whenever α is a limit point of C, C ∩ α = Cα.

Furthermore, Sδ ∗ Tλδ has a λ-closed dense subset of size 2δ.

Theorem 5.16. Suppose κ is super-almost-huge and µ < κ is regular. Then there is a

µ-distributive forcing extension in which κ = µ+, �λ holds for all λ ≥ κ, and for all regular

λ ≥ κ there is a normal, fine, κ-complete, λ-dense ideal on Pκ(λ).

Proof. By Chapter 2, we may pass to a µ-distributive forcing extension in which κ = µ+

and for all regular λ ≥ κ there is a normal, fine, κ-complete, λ-dense ideal on Pκ(λ), and

GCH holds above µ. Over this model, force with P, the Easton support product of Sλ where

λ ranges over all cardinals ≥ κ. For every cardinal λ, P naturally factors into P<λ × P≥λ.

Note that if λ ≥ κ, P≥λ is (λ+ 1)-strategically closed.

First we show that for each regular λ ≥ κ, P≥λ is λ+-distributive in V P<λ . Suppose that

H0 ×H1 is (P<λ × P≥λ)-generic, and f : λ → Ord is in V [H0][H1]. Then in V [H1], there is

a P<λ-name τ for f . By GCH and the fact that we take Easton support, |P<λ| = λ, so it is

λ+-c.c. in V [H1]. Thus τ may be assumed to be a subset of V of size λ. By the strategic

closure of P≥λ, τ ∈ V . Thus f = τH0 ∈ V [H0], establishing the claim.

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Next we show that P preserves all regular cardinals. First note that since P is (κ + 1)-

strategically closed, P cannot change the cofinality of any regular δ to some λ ≤ κ. If P does

not preserve regular cardinals, then in some generic extension V [G], there are λ < δ which

are regular in V with κ < λ, such that V [G] |= cf(δ) = λ. Let H = H0×H1, where H0 ⊆ P<λ

and H1 ⊆ P≥λ. By the λ+-c.c. of P<λ, V [H0] |= cf(δ) > λ, and by the λ+-distributivity of

P≥λ in V [H0], V [H] |= cf(δ) > λ, a contradiction. Since a square sequence is upwards

absolute to models with the same cardinals and Sλ regularly embeds into P for all λ ≥ κ, P

forces (∀λ ≥ κ)�λ.

For each regular λ ≥ κ, let Zλ = Pκ(λ). We want to show that in V P, for each regular λ ≥ κ,

there is a normal, fine, λ-dense ideal on Zλ. It suffices to show that such an ideal exists in

V P<λ , since P≥λ adds no subsets of λ, and |Zλ| = λ. First note that by the strategic closure

of P, the dense ideal on κ is unaffected.

Let Q be the Easton support product of Sλ ∗ Tµλ, where λ ranges over all cardinals. There

is a coordinate-wise regular embedding of P into Q. When λ is regular, Q<λ has a dense

µ-closed subset of size λ. Hence it regularly embeds into B(Col(µ, λ)). The dense ideal Iλ

on Zλ in V has quotient algebra isomorphic to B(R × Col(µ, λ)) for some small R, and so

Q<λ regularly embeds into this forcing.

If G ⊆ P(Zλ)/Iλ is generic, let H be the induced generic for P<λ, and let j : V →M ⊆ V [G]

be the ultrapower embedding. Recall that crit(j) = κ, j(κ) = λ+, λ++ is a fixed point of j,

and j[λ] ∈M . First note that j[λ] \ j(κ) is an Easton set in M . If j(κ) ≤ δ ≤ j(λ) and δ is

regular in M , then since ot(j[λ] ∩ δ) ≤ λ < δ, sup(j[λ] ∩ δ) < δ.

For each cardinal δ such that κ ≤ δ < λ, let 〈Cδα : α < δ+〉 be the �δ sequence and let tδ be

the “thread” of order type µ, both given by given by H � (Sδ ∗ Tµδ ). By the µ-distributivity

of Sδ ∗ Tµδ , all initial segments of tδ are in V , and since they are small, j(tδ ∩ α) = j[tδ ∩ α]

for α < δ+, and j is continuous at all limit points of tδ. Let γδ = sup(j[δ+]) < j(δ+), and in

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M consider mδ =⋃α<δ+ j(〈Cδ

α : β < α〉) ∪ {(γδ, j[tδ])}. Each mδ is a condition in (Sj(δ))M ,

and the sequence m = 〈mδ : δ ∈ j[λ] \ κ ∩ CardM〉 is a condition in (P<j(λ))M .

M thinks P<j(λ) is j(κ)-strategically closed, and this is true in V [G] as well since these models

share the same λ-sequences, and j(κ) = λ+ in V [G]. Since j(λ+) < (λ++)V , P(P<j(λ))M has

cardinality λ+ in V [G]. Thus we may use the winning strategy to build a filter H ⊆ (P<j(λ))M

that is generic over M , with m ∈ H. Since m is a lower bound to j(p) for all p ∈ H, we

have j[H] ⊆ H.

Therefore, the hypotheses of Theorem 5.1 are satisfied, with respect to Zλ, Iλ,B(P<λ). No

further forcing over V [G] is required to build H, so the ideal K from Theorem 5.1 is prime.

Thus we have a P<λ-name for a normal and fine ideal Jλ on Zλ such that B(P<λ∗ ˙P(Zλ)/Jλ) ∼=

B(P(Zλ)/Iλ). Hence Jλ is λ-dense in V [H].

We note that in the case κ = ω1, P(Zλ)/Jλ ∼= B(Col(ω, λ)) for all regular λ. But for

higher cardinals, Proposition 3.8 shows the quotient algebras must differ from those given by

Theorem 2.17. The crux is that the “threading” forcings are left over as regular suborders.

It is not possible to improve this result to get the consistency of, “For all cardinals λ ≥ κ,

�λ and there is a normal, fine, λ-dense ideal on Pκ(λ).” Burke and Matsubara [5] showed

that if cf(λ) < κ and there is normal, fine, κ-complete, λ+-saturated ideal on Pκ(λ), then

every stationary subset of λ+ ∩ cof(<κ) reflects.

5.4 Mutual inconsistency

In contrast to the observed situation with traditional large cardinals, generic large cardinals

can be individually consistent yet mutually inconsistent. Only a handful of examples of this

phenomenon seem to be known; this is discussed in Section 11.2 of [13]. Here, we use our

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“destruction” method to bring more instances to light.

Theorem 5.17. Let µ+ = κ < δ be regular cardinals. The following are mutually inconsis-

tent:

(1) There is a κ-complete, κ-dense ideal I on κ.

(2) There is a normal, fine, κ-complete, δ-saturated ideal J on Z = [δ]κ, such that P(Z)/J

is δ-absolutely δ-c.c. and d(P(A)/J) = δ for all A ∈ J+.

Proof. Suppose the two statements hold simultaneously. We may assume that I is a normal

ideal on κ. Let G be generic for P(Z)/J and let j : V → M ⊆ V [G] be the associated

embedding. Since J is κ-complete, crit(j) ≥ κ, and since j[δ] ∈ j(Z), |δ| = j(κ) in M , so

crit(j) = κ. Since the forcing to produce G is δ-c.c., j(κ) = δ. Thus by elementarity, M |=

“There is a normal, fine, δ-dense ideal on δ.” By the closure properties of M , P(δ)V [G] =

P(δ)M , so there is such an ideal in V [G]. But by Lemma 5.10, forcing with P(Z)/J destroys

all dense ideals on δ.

This theorem strengthens and generalizes a result of Woodin. As described in Section 5.6

of [13], Woodin used some results of Laver and Hajnal-Juhasz on partition properties to

show that the following are mutually inconsistent:

(1) There is a countably complete, ω1-dense ideal on ω1.

(2) There is a normal and fine ideal J on Z = [ω2]ω1 such that P(Z)/J ∼= B(Col(ω,< ω2)).

(3) CH.

In Theorem 5.17, we get to eliminate CH from the list, and speak about a broader class

of partial orders in statement (2). It is easy to see that Col(ω,< ω2) has uniform density

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ω2 and is ω2-absolutely ω2-c.c. Furthermore, Theorem 5.17 applies to more pairs of regular

cardinals κ < δ.

In Chapter 2, we saw that statement (1) in Theorem 5.17 is consistent relative to an almost-

huge cardinal. It remains open whether there can be an ideal as in statement (2) when

δ = κ+, but known results give models of (2) with δ > κ+. Starting from a huge cardinal,

Magidor [29] produced a forcing extension in which there is a normal, fine, ω3-saturated ideal

on [ω3]ω1 . Huberich [17] improved upon this, producing ideals on more general spaces having

stronger saturation properties. He started with a model of GCH, µ < κ < λ < δ regular

cardinals, and κ carrying a huge embedding with target δ. He showed that there is a forcing

extension with the same cardinals in [0, µ] ∪ [κ, λ], and in which κ = µ+, δ = λ+, and there

is a normal, fine, κ-complete, λ-centered ideal J on Z = [δ]κ. This means that P(Z)/J is

the union of λ many filters, and it implies that P(Z)/J is δ-c.c. in any outer model in which

δ remains a cardinal. An elementarity argument shows that any λ-centered partial order is

δ-absolutely δ-c.c. Huberich showed that P(Z)/J is isomorphic to a complete subalgebra

of a certain complete boolean algebra B, and it is easy to check that d(B � b) = δ for all

b ∈ B. However, one can show using arguments similar to those following Theorem 2.17 that

P(Z)/J ∼= B in this model. Therefore, statement (2) of Theorem 5.17 is consistent relative

to a huge cardinal.

If we modify statement (2) to require that the ideal J has density less than δ, there is a

chance it is consistent with (1). The question of whether there can be a normal, fine, ω1-

dense ideal on [ω2]ω1 was raised by Foreman. We note that if this is possible, it would solve

the question discussed at the end of Chapter 2.

Proposition 5.18. Suppose there is a normal, fine, κ-complete, κ-dense ideal on [κ+]κ.

Then there are dense ideals on κ and κ+.

Proof. Let J be a κ-complete, κ-dense ideal on Z = [κ+]κ. Let I = {X ⊆ κ : 1 P(Z)/I κ /∈

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j(X)}. As in the proof of Corollary 3.15, I is normal, fine, and κ-dense. If G ⊆ P(Z)/J

is generic, then in V [G] there is a normal, fine, (κ+)V -dense ideal K on (κ+)V . In V , let

P = P(Z)/J ∗P(κ+)/K, and let L = {X ⊆ κ+ : 1 P κ+ /∈ k(X)}, where k is the elementary

embedding associated to K. Also as in Corollary 3.15, L is normal, fine, κ++-saturated, and

there is a complete embedding of P(κ+)/L into B(P), so L is κ+-dense.

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Chapter 6

Coherent forests

A question is immediately raised by the argument for Theorem 5.11. Can we obtain the

same consistency result simultaneous with GCH? In particular, does CH + a nonregular

ideal on ω2 imply the existence of a dense ideal on ω2? We may assume we start with a

model of GCH plus normal, fine, κ-complete, λ-dense ideals on Pκ(λ), where κ = µ+. Does

there exist a µ-c.c. partial order of uniform density κ that preserves GCH below κ? Typical

µ-c.c., uniformly κ-dense forcings will introduce κ many subsets of some cardinal ν < µ. If

we wish to preserve µ<µ = µ, we will need a special kind of forcing.

At first glance, we may conjecture that such objects cannot exist alongside dense ideals. Per-

haps the assumption of µ-strategic closure in Lemma 4.5 can be weakened to µ-distributivity.

Combined with Proposition 3.6, this would imply that the existence of a κ-complete, κ-dense

ideal on κ is equivalent to the existence of a κ-complete nonregular ideal on κ when κ = µ+,

µ is regular, and µ<µ = µ. But perhaps not.

Suppose µ<µ = µ, and P is a µ-c.c. forcing that preserves this. We can take a set of P-names

{τα : α < µ} whose realizations are forced to become Pµ(µ). By the µ-c.c., we can assume

that each |τα| < µ. If θ > κ is regular such that P ∈ Hθ, then we can take an elementary

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substructure M ≺ Hθ such that P, {τα : α < µ} ∈ M , |M | = µ, and M<µ ⊆ M . Then

P0 = P ∩M is a regular suborder of P. We have some P0-name Q such that P ∼= P0 ∗ Q,

and P0 B(Q) is (µ, µ)-distributive. By the µ-c.c. of Q in V P0 , Q is thus forced to be µ-

distributive. Hence, B(Q) is, by definition, a µ-Suslin algebra in V P0 . Since |P0| < κ, any

κ-complete, λ-dense ideal will remain so after forcing with P0.

Therefore, if we wish to apply the technique of Chapter 5 to destroy all normal, fine, κ-

complete, λ-dense ideals on Pκ(λ) while preserving a nonregular one, we will need a model

in which there is such a dense ideal and also a µ-Suslin algebra of uniform density κ. Such a

model is produced in this chapter. We also include several related results that were obtained

in the pursuit of this model.

We consider a type of structure called a forest, a generalization of a tree. Forests contain

many trees, but can be much wider than a single tree. Thomas Jech had previously studied

the same type of object under the name “mess” [19]. The nicer choice of terminology is due

to Christoph Weiß [38]. In contrast to the work of Weiß, we will focus on forests that do not

contain long branches.

Definition. A (κ,X, µ)-forest is a collection of functions F satisfying:

(1) {dom(f) : f ∈ F} = Pκ(X).

(2) (∀f ∈ F ) ran(f) ⊆ µ.

(3) For z ∈ Pκ(X), let Fz = {f ∈ F : dom(f) = z}. A forest must satisfy that for z0 ⊆ z1

in Pκ(X), Fz0 = {f � z0 : f ∈ Fz1}.

Forests are full of trees. If F is a (κ,X, µ)-forest, and S = {xα : α < κ} is an enumeration

of distinct elements of X, then TS = {f ∈ F : (∃β < κ) dom(f) = {xα : α < β}} forms a

tree of height κ under the subset ordering.

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A (κ,X, µ)-forest F is called thin if for all z ∈ Pκ(X), |Fz| < κ. A collection of functions F

is called κ-coherent if for all f, g ∈ F , |{x ∈ dom(f) ∩ dom(g) : f(x) 6= g(x)}| < κ. If F is a

(κ+, X, µ)-forest we say it is coherent if it is κ-coherent. Clearly, if µ ≤ κ = κ<κ, then any

coherent (κ+, X, µ)-forest is thin.

A chain in a forest is a subset which is linearly ordered under ⊆. Two elements f, g in a

forest F are said to be compatible when they have a common extension h ∈ F . An antichain

in a forest is a subset of pairwise incompatible elements. We say that a (κ,X, µ)-forest

F is Aronszajn if it contains no well-ordered chain of length κ. We say it is Suslin if it

contains no antichain of cardinality κ. If F is a (κ,X, µ)-forest with µ ≥ 2, closed under

finite modifications, then F is Suslin only if it is Aronszajn. This is because we can “split

off” from any chain of length κ to get an antichain of size κ.

Proposition 6.1. If F is a (κ,X, µ)-forest, then for any z ∈ Pκ(X), Fz is a maximal

antichain.

Proof. Let f ∈ F , z ∈ Pκ(X). By clause (3) of the definition of forests, there is g ∈ F such

that f ⊆ g and dom(g) = dom(f) ∪ z. Then g � z ∈ Fz, so g is a common extension of f

and something in Fz.

The following lemma will be useful in several constructions:

Lemma 6.2. Suppose F is a coherent (κ+, X, µ)-forest, and F is closed under <κ modifi-

cations. Then two functions in F have a common extension in F if and only if they agree

on their common domain.

Proof. Let f, g ∈ F agree on dom(f)∩ dom(g). Let h ∈ F be such that dom(h) = dom(f)∪

dom(g). By coherence, we can change the values of h on a set of size <κ to get h′ : dom(h)→

µ with h′ � dom(f) = f , and h′ � dom(g) = g. By the closure of F , h′ ∈ F .

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6.1 Aronszajn forests

The first theorem of this section generalizes of an argument of Koszmider [24].

Lemma 6.3. Let κ be a regular cardinal, and suppose F = {fα : α < κ} is a κ-coherent set

of partial functions from κ to µ.

(a) There is a function f : κ→ µ such that {f} ∪ F is κ-coherent.

(b) If µ = κ and each fα is <κ to 1, then there is a <κ to 1 function f : κ → κ such that

{f} ∪ F is κ-coherent.

Proof. For each α, let Dα = dom(fα) \⋃β<α dom(fβ). Let E = κ \

⋃αDα. For the first

claim, choose any function g : E → µ, and let

f(β) =

fα(β) if β ∈ Dα

g(β) if β ∈ E

For any α, {β : f(β) 6= fα(β)} =⋃γ<α{β ∈ Dγ ∩ dom(fα) : fγ(β) 6= fα(β)}. This is a union

of <κ sets of size <κ, so has size <κ.

For the second claim, choose any <κ to 1 function g : E → κ, and let

f(β) =

max(α, fα(β)) if β ∈ Dα

g(β) if β ∈ E

For any α, {β : f(β) 6= fα(β)} ⊆⋃γ≤α{β ∈ Dγ : fγ(β) < γ or fγ(β) 6= fα(β)}. By the

hypotheses, this set has size <κ. For each α, f−1(α) ⊆ g−1(α)∪⋃{f−1

γ (β) : γ, β ≤ α}, so f

is <κ to 1.

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Theorem 6.4. Let κ be a regular cardinal. For every ζ < κ, there is a coherent (κ+, κ+ζ , κ)-

forest consisting of <κ to 1 functions.

Proof. We will prove by induction the following stronger statement: For every ζ < κ and

every sequence 〈(Xα, Fα) : α < κ〉 such that:

(1) each Xα ⊆ κ+ζ ,

(2) each Fα is a (κ+, Xα, κ)-forest of <κ to 1 functions,

(3)⋃α Fα is κ-coherent,

there is a coherent (κ+, κ+ζ , κ)-forest F ⊇⋃α Fα consisting of <κ to 1 functions.

For ζ = 0, pick a collection {fα : α < κ} such that for each α, fα ∈ Fα, and dom(fα) = Xα.

By Lemma 6.3(b), there is a < κ to 1 function f : κ→ κ that coheres with each fα, and we

can take F = {g : dom(g) ⊆ κ and |{x : f(x) 6= g(x)}| < κ}.

Assume ζ = η+ 1 and the statement holds for η. For each β < κ+ζ , let F βα =

⋃α{f � β : f ∈

Fα}. We will construct F ⊇⋃Fα as the union of a ⊆-increasing sequence 〈Gβ : β < κ+ζ〉

such that for each β, Gβ is a coherent (κ+, β, κ)-forest of <κ to 1 functions containing⋃α F

βα .

Let G0 = {∅}. Given Gβ, let Gβ+1 = {f : dom(f) ⊆ (β + 1), ran(f) ⊆ κ, and f � β ∈ Gβ}.

Suppose β is a limit ordinal of cofinality ≤ κ, and let 〈γi : i < δ ≤ κ〉 be cofinal in β. The

collection⋃i<δ Gγi ∪

⋃α<κ F

βα is κ-coherent, because (∀α < κ)(∀f ∈ F β

α )(∀i < δ)(f � γi ∈

F γiα ⊆ Gγi). Since β has cardinality ≤ κ+η, the inductive assumption implies that we can

extend to a forest Gβ with the desired properties.

Suppose β is a limit ordinal of cofinality > κ. Let Gβ =⋃γ<β Gγ. Then Gβ is a forest with

the desired properties because⋃α<κ F

βα =

⋃γ<β(

⋃α<κ F

γα ). Finally, we let F =

⋃β<κ+ζ Gβ.

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Now assume ζ is a limit ordinal of cofinality < κ, and the statement holds for all η < ζ.

Let 〈γi : i < δ = cf(ζ)〉 be an increasing cofinal sequence in ζ. Like above, recursively build

an increasing sequence 〈Gi : i < δ〉 such that each Gi is a (κ+, κ+γi , κ)-forest of < κ to 1

functions extending⋃α F

γiα . This is done by applying the inductive hypothesis for κ+γi to⋃

α Fγiα ∪

⋃j<iGj. We may also assume each Gi is closed under < κ modifications. Simply

let F be the collection of functions f such that dom(f) ⊆ κ+ζ , and (∀i < δ)f � γi ∈ Gγi .

Clearly F ⊇⋃α Fα.

First note that if f ∈ F were not < κ to 1, then there would be some i < δ such that

f � κ+γi is not < κ to 1, which is false. If f, g ∈ F were to disagree at κ many points,

then there would be some i < δ such that f � κ+γi and g � κ+γi disagree at κ many points,

which is false. Second, we check that for any z ∈ Pκ+(κ+ζ), there is an f ∈ F such that

dom(f) = z. We can recursively build a sequence 〈gi : i < δ〉 such that for all i < j < δ,

gi ∈ Gi, dom(gi) = z ∩ κ+γi , and gi ⊆ gj. If we have built such a sequence up to j < δ, then⋃i<j gi ∈ Gj, because for any h ∈ Gj with domain z ∩ κ+γj , the set of disagreements with⋃i<j gi has size < κ. Let f =

⋃i<δ gi.

Koszmider showed that in the case κ = ω, if λ is a singular cardinal of cofinality ω, and �λ

and λω = λ+ hold, then the induction can push through λ as well. The argument generalizes

almost verbatim to show for any regular κ, the induction can go forward at λ of cofinality

κ, under the assumptions �λ and λκ = λ+. As a consequence, we get that in L, for every

regular κ and every λ ≥ κ, there is a coherent, (κ+, λ, κ)-forest of < κ to 1 functions.

Recall that a partial order P is called κ-Knaster if for any A ⊆ P of size κ, there is B ⊆ A

of size κ that consists of pairwise compatible elements.

Corollary 6.5. For every regular cardinal κ and every ζ < κ, there is a coherent (κ+, κ+ζ , κ)-

forest, which is Aronszajn, does not have the 2<κ or the κ+ chain condition, but is (2κ)+-

Knaster. If ζ is finite or 2<κ < κ+ω, then the forest is (2<κ · κ+)+-Knaster.

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Proof. Let F be given by Theorem 6.4. We may assume F is closed under <κ modifications.

To see the failure of the 2<κ chain condition, note that for any z ⊆ κ+ζ of size κ, Fz is an

antichain of size 2<κ.

Let {αβ : β < κ+} be any enumeration of distinct ordinals in κ+ζ , and for each γ < κ+,

let fγ ∈ F have domain {αβ : β < γ}. Since each f ∈ F maps into κ, there is a ξ < κ

and a stationary subset S0 ⊆ {γ < κ+ : cf(γ) = κ} such that for all γ ∈ S0, fγ+1(αγ) = ξ.

Since each f ∈ F is < κ to 1, each set {β < γ : fγ+1(αβ) = ξ} is bounded below γ when

cf(γ) = κ. Thus there is an η < κ+ and a stationary S1 ⊆ S0 such that for all γ ∈ S1,

{β < γ : fγ+1(αβ) = ξ} ⊆ η. Therefore, for any γ0 < γ1 in S1 \ η, fγ0+1(αγ0) 6= fγ1+1(αγ0).

This shows that F does not have the κ+ chain condition.

It also shows that F is Aronszajn. For otherwise, let 〈fα : α < κ+〉 be a strictly increasing ⊆-

chain in F . Let {ξβ : β < κ+} =⋃α dom(fα), and for each γ let gγ = (

⋃α fα) � {ξβ : β < γ}.

Then 〈gγ : γ < κ+〉 is a strictly increasing chain, but by the above paragraph, it contains an

antichain of size κ+, contradiction.

To show the (2κ)+-Knaster property, let {fα : α < (2κ)+} ⊆ F . Let T0 ⊆ (2κ)+ have size

(2κ)+ and be such that {dom(fα) : α ∈ T0} forms a delta-system with root r. Let T1 ⊆ T0

have size (2κ)+ and be such that for a fixed g, fα � r = g for all α ∈ T1. The union of any

two of these is in F .

For the case where ζ < ω or 2<κ < κ+ω, let θ = (2<κ · κ+)+. First note that it is easy

to see by induction that for every n < ω, Pκ+(κ+n) has a cofinal subset of size κ+n. Let

A = {fα : α < θ} ⊆ F , and let S =⋃α dom(fα).

Suppose first that |S| < θ. There is an R ⊆ Pκ+(S) that covers {dom(fα) : α < θ} and

has cardinality |S|. Therefore, by the coherence of F , there is a G ⊆ F of cardinality

≤ |S| · 2<κ < θ such that for all α < θ, there is g ∈ G with fα ⊆ g. Therefore there is a

g0 ∈ G which is a common lower bound to θ many fα.

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Now suppose that |S| = θ. Since θ is regular and θ > κ+, we can use the delta-system

argument to get an S0 ⊆ S of cardinality less than θ and a T0 ⊆ θ of cardinality θ such that

for all α0, α1 ∈ T0, dom(fα0) ∩ dom(fα1) ⊆ S0. By the above paragraph, there is a T1 ⊆ T0

of cardinality θ such that for any α0, α1 ∈ T1, fα0 and fα1 agree on their common domain

contained in S0.

One may ask whether the condition “< κ to 1” can be strengthened to “1 to 1” in Theo-

rem 6.4. But this cannot always be achieved:

Proposition 6.6. If there is a coherent (κ+, λ, κ)-forest consisting of injective functions,

then there are λ many almost disjoint subsets of κ.

Proof. Let F be such a forest, and for each z ∈ Pκ+(λ), choose fz ∈ F with domain z. Let

S be a collection of λ many pairwise disjoint subsets of λ, each of cardinality κ. For x 6= y

in S, ran(fx) is almost disjoint from ran(fy). This is because the sets A = ran(fx∪y � x) and

B = ran(fx∪y � y) are disjoint, and |A4 ran(fx)| < κ, and |B4 ran(fy)| < κ.

A positive answer in the following special case is well-known (see [25], Chapter II, Theorem

5.9 and exercise 37):

Theorem 6.7. Let κ be a regular cardinal. There is a κ-coherent collection of functions

{fα : α < κ+}, such that each fα is an injection from α to κ.

A more general positive answer can be forced:

Theorem 6.8. Assume κ is a regular cardinal with 2<κ = κ, and λ ≥ κ. There is a κ-closed,

κ+-c.c. partial order that adds a coherent (κ+, λ, κ)-forest of injective functions.

Proof. Let P be the collection partial functions p that assign to < κ many z ⊆ λ of size ≤ κ,

a partial injective function from z to κ defined at < κ many points. Let p ≤ q when:

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(a) dom(p) ⊇ dom(q).

(b) For all z ∈ dom(q), p(z) ⊇ q(z).

(c) If z0, z1 ∈ dom(q), α ∈ z0 ∩ z1 \ (dom(q(z0)) ∪ dom(q(z1)), and α ∈ dom(p(z0)), then

α ∈ dom(p(z1)) and p(z0)(α) = p(z1)(α).

It is easy to check that ≤ is transitive and that 〈P,≤〉 is κ-closed. To check the chain

condition, let A ⊆ P have size κ+. Since κ<κ = κ, we can find a B ⊆ A of size κ+ such that

{dom(p) : p ∈ B} forms a delta-system with root R. Again since κ<κ = κ, there is a C ⊆ B

of size κ+ and a collection of functions {fz : z ∈ R} such that ∀p ∈ C, ∀z ∈ R, p(z) = fz. If

p, q ∈ C, then p ∪ q is a common extension.

If G ⊆ P is generic, then for all z ∈ Pκ+(λ)V , G gives an injective function fz : z → κ as⋃{p(z) : z ∈ p ∈ G}. For z0, z1 ∈ Pκ+(λ)V , there is some p ∈ G such that z0, z1 ∈ dom(p). p

forces that fz0 and fz1 agree outside dom(p(z0))∪dom(p(z1)). Finally, by the κ+-c.c., Pκ+(λ)V

is cofinal in Pκ+(λ)V [G]. So we can define a (κ+, λ, κ)-forest F as {f : f is an injection into

κ, (∃z) dom(f) ⊆ z ∈ Pκ+(λ)V , and f disagrees with fz at < κ many points}.

Such a forest will be Aronszajn because a chain of length κ+ would give an injection from

κ+ to κ. Unlike the forests of Theorem 6.4, it will never have the λ chain condition.

6.2 Influence of the P-ideal dichotomy

In the previous section, we saw that coherent, Aronszajn (ω1, ωn, ω)-forests can be con-

structed in ZFC for every natural number n. Here we show that the third coordinate is

optimal, in the sense that for n < ω and λ ≥ ω1, ZFC cannot prove the existence of a

coherent, Aronszajn (ω1, λ, n)-forest. Let us recall the relevant notions:

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Definition. An ideal I ⊆ P(X) is a P-ideal when Pω(X) ⊆ I ⊆ Pω1(X), and for any

{zn : n < ω} ⊆ I, there is z ∈ I such that zn \ z is finite for all n.

Definition. The P-ideal dichotomy (PID) is the statement that for any P-ideal I on a set

X, either

(1) there is an uncountable Y ⊆ X such that Pω1(Y ) ⊆ I, or

(2) there is a partition of X into {Xn : n < ω} such that for all n and all z ∈ I, z ∩Xn is

finite.

PID is a consequence of the Proper Forcing Axiom, and is also known to be consistent with

ZFC+GCH relative to a supercompact cardinal [36]. The restriction of PID to ideals on

sets of size ω1 is known to be consistent without the use of large cardinals, both with and

without GCH [1].

Using a coherent, Aronszajn (ω1, ω1, ω)-forest F , we can obtain a coherent, Aronszajn ω1-

tree T of binary functions by taking the collection of characteristic functions of members of

F whose domain is an ordinal, considering the functions as subsets of α × ω for α < ω1. A

cofinal branch would be a function g : ω1 × ω → 2 with g � (α× ω) ∈ T for all α < ω1, and

this would code an uncountable well-ordered chain in F . Further, using a regressive function

argument, we can see that the closure of T under finite modifications remains Aronszajn.

On the other hand, forests are more flexible. If we take such a tree T , close it under subsets

to get a forest F , then it may be that there is an uncountable well-ordered chain C ⊆ F ,

but with dom(⋃C) a proper subset of ω1 × ω. This is what happens under PID.

Theorem 6.9. Assume PID, and let F be a coherent (ω1, λ, n)-forest closed under finite

modifications, for some λ ≥ ω1, n < ω. Then F is not Aronszajn.

Proof. First we prove this for n = 2. Let F be a coherent (ω1, λ, 2)-forest closed under finite

modifications. Let I be the collection of z ⊆ λ such that for some f ∈ F , z ⊆ {α : f(α) = 1}.

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We claim I is a P-ideal. Let {zn : n < ω} ⊆ I, and for each n, choose fn ∈ F witnessing

zn ∈ I. Let f ∈ F have domain⋃n dom(fn), and let z = {α : f(α) = 1}. For any n, f

disagrees with fn on a finite set, so there can only be finitely many α ∈ zn \ z.

Assume that alternative (1) of PID holds, and let Y ⊆ λ be uncountable such that Pω1(Y ) ⊆

I. Enumerate Y as 〈yα : α < ω1〉. For each α < ω1, let fα be the function that has fα(yβ) = 1

for β < α, and is undefined elsewhere. Since F is closed under subsets, each fα ∈ F , and

these form an uncountable well-ordered chain.

Assume alternative (2) of PID holds. Let Xn ⊆ λ be uncountable such that for all z ∈ I,

Xn ∩ z is finite. Let g have constant value 0 on Xn. If f ∈ F and dom(f) ⊆ Xn, then

{α : f(α) = 1} is finite. Thus for any countable z ⊆ Xn, g � z ∈ F , so again we have an

uncountable well-ordered chain.

Now assume the result holds for n, and let F be a coherent (ω1, λ, n+1)-forest. Let r(k) = 0

for k < n, and r(n) = 1. Consider the forest G = {r ◦ f : f ∈ F}, and let g0, g1 be the

functions on λ with constant value 0 and 1 respectively. By the above argument, there is

some uncountable Y ⊆ λ such that either g0 � z ∈ G for all countable z ⊆ Y , or likewise

for g1. The latter case shows that F is not Aronszajn. In the former case, we have that

for all countable z ⊆ Y , there is a function fz ∈ F with domain z that only takes values

below n. If H = {g : (∃z ∈ Pω1(Y ))g : z → n and {α : g(α) 6= fz(α)} is finite}, then

H is a coherent (ω1, Y, n)-forest contained in F . By induction, H contains an uncountable

well-ordered chain.

6.3 Suslin forests

Lemma 6.10. Let κ be a regular cardinal. All Suslin (κ, λ, µ)-forests are κ-distributive.

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Proof. Let F be a Suslin (κ, λ, µ)-forest, and let 〈Aα : α < δ < κ〉 be a sequence of

maximal antichains contained in F . By the Suslin property, each Aα has size < κ, so if

z =⋃α{dom(f) : f ∈ Aα}, |z| < κ. By maximality, for every α < δ and every g ∈ Fz, there

is an f ∈ Aα such that g is compatible with f . But since dom(f) ⊆ dom(g), this means

f ⊆ g. Thus Fz refines each Aα.

The boolean completion of a Suslin (κ, λ, µ)-forest is a κ-Suslin algebra, which is a complete

boolean algebra with that is both κ-c.c. and κ-distributive. The cardinality of this algebra is

at least λ. Therefore the existence of varieties Suslin forests is constrained by the following

(see [21], Theorem 30.20):

Theorem 6.11 (Solovay). If B is a κ-Suslin algebra, then |B| ≤ 2κ.

Large Suslin forests can be obtained by forcing. In [19], Jech defined a class of partial orders

Pλ such that under CH, Pλ is countably closed, ω2-c.c., and adds a Suslin (ω1, λ, 2)-forest.

However, this forest fails to be coherent. Modifying his forcing slightly, we obtain:

Theorem 6.12. Assume κ is a regular cardinal, 2<κ = κ, and 2κ = κ+. Then for all λ > κ,

there is a κ+-closed, κ++-c.c. forcing of size λ<κ that adds a coherent, Suslin (κ+, λ, 2)-forest.

Proof (sketch). Let P be the set of all partial functions f from λ to 2 of size ≤ κ, and say

f ≤ g when dom(f) ⊇ dom(g) and |{α : f(α) 6= g(α)}| < κ. κ+-closure follows from

Lemma 6.3(a), and the κ++-c.c. follows from a delta-system argument. If G is P-generic

over V , in V [G] let F = {f : (∃g ∈ G) dom(g) = dom(f) and |{α : f(α) 6= g(α)}| < κ}.

Clearly F is coherent. The argument that F is Suslin in V [G] proceeds as in [19].

By adapting an argument of Todorcevic that appears in [35], we can obtain large Suslin

forests in a different way:

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Theorem 6.13. Assume κ is a regular cardinal, 2<κ = κ, and there is a coherent (κ+, λ, κ)-

forest of injective functions. Then adding a Cohen subset of κ adds a coherent, Suslin

(κ+, λ, 2)-forest.

Proof. Let F be a coherent (κ+, λ, κ)-forest of injections closed under < κ modifications to

other injections. Let g : κ → 2 be an Add(κ) generic function over V . Consider the family

G0 = {g ◦ f : f ∈ F}. Since Add(κ) is κ+-c.c., Pκ+(λ)V is cofinal in Pκ+(λ)V [g], so G0

generates a forest G when we close under subsets. G inherits coherence from F . We claim

G is Suslin.

First we note that G is closed under < κ modifications. If f ∈ F , then by the argument for

Proposition 6.6, κ\ran(f) has size κ. By a density argument, {α ∈ κ\ran(f) : g(α) = i} has

size κ for both i = 0, 1. So if g ◦ f ∈ G, and x ⊆ dom f has size < κ, we can switch values of

g◦f on x by choosing distinct ordinals {αi : i ∈ x} ⊆ κ\ran(f) such that g(αi) = g(f(i))+1

mod 2. If f ′ = f except that f ′(i) = αi for i ∈ x, then f ′ ∈ V by κ-closure, so g ◦ f ′ ∈ G. So

by Lemma 6.2, members of G have a common extension when they agree on their common

domain.

Towards a contradiction, suppose A = {g ◦ fα : α < κ+} is an antichain in G0, and let

p0 ∈ Add(κ) force this. Since |Add(κ)| = κ, there is some p1 ≤ p0 such that p1 g ◦ f ∈ A

for κ+ many f ∈ F . Let A0 = {f : p1 g ◦ f ∈ A}, and let Z =⋃{dom(f) : f ∈ A0}.

Case 1: |Z| ≤ κ. Let h ∈ F be such that dom(h) = Z. There are at most κ many < κ

modifications of h, so there are f0, f1 ∈ A0 such that both agree with the same modification

of h. But p1 forces that g ◦ f0 and g ◦ f1 are compatible, contradiction.

Case 2: |Z| = κ+. Let 〈αi : i < κ+〉 be an enumeration of Z. Let β0 = sup(dom(p1)) + 1,

and for each f ∈ A0, let Xf = {α : f(α) < β0}. Since each f is injective, each |Xf | < κ. For

each Xf , let 〈Xf (i) : i < βf〉 be an enumeration of Xf that agrees in order with the above

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enumeration of Z.

Case 2a: There is no i < κ such that |{Xf (i) : f ∈ A0}| = κ+. Then there is a γ < κ+ such

that for all f ∈ A0, {i : αi ∈ Xf} ⊆ γ. Since κ<κ = κ, we may choose some A1 ⊆ A0 such

that for all f ∈ A1, Xf is the same set S, and further that f � S is the same for all f ∈ A1.

Let f0, f1 ∈ A1, and let D = {α ∈ dom(f0) ∩ dom(f1) : f0(α) 6= f1(α)}. |D| < κ, D ∩ S = ∅,

and if α ∈ D, then f0(α), f1(α) ≥ β0. Thus we can define a q ≤ p1 such that for all α ∈ D,

q ◦ f0(α) = q ◦ f1(α) = 0. q forces that g ◦ f0 and g ◦ f1 are compatible, contradiction.

Case 2b: There is some i < κ such that |{Xf (i) : f ∈ A0}| = κ+. Let i0 be the least such

ordinal. We choose a sequence 〈fα : α < κ+〉. Let f0 ∈ A0 be arbitrary. Let f1 be such that

Xf1(i0) has index in the enumeration of Z above {i : αi ∈ dom(f0)}. Keep going in this

fashion such that for β < γ < κ+, Xfγ (i0) has index greater than sup{i : αi ∈ dom(fβ)}. By

the minimality of i0, there is C ⊆ κ+ of size κ+ and a set S ⊆ Z such that for all α ∈ C,

{Xfα(i) : i < i0} = S, and fα � S is the same.

Now let β < γ be in C, and let D = {α ∈ dom(fβ) ∩ dom(fγ) : fβ(α) 6= fγ(α)}. As before,

|D| < κ and D∩S = ∅. If α ∈ D, then fγ(α) ≥ β0, because Xfγ∩dom(fβ) = S. We construct

q ≤ p1 such that for all α ∈ D, q ◦ fγ(α) = q ◦ fβ(α). Let D0 = {α ∈ D : fβ(α) ∈ dom(p1)},

and let q0 = p1∪{〈fγ(α), p1 ◦ fβ(α)〉 : α ∈ D0}. We are free to do this because fγ is injective

and fγ(α) /∈ dom(p1) for α ∈ D.

Note that for all α ∈ D, q0 is defined at fγ(α), only if it is defined at fβ(α). But it may be

that for some α ∈ D0 and some α′ ∈ D \ D0, fγ(α) = fβ(α′). Assume we have a sequence

q0 ≥ ... ≥ qn such that:

(1) for all k ≤ n, D ∩ f−1γ [dom(qk)] ⊆ D ∩ f−1

β [dom(qk)],

(2) for all k ≤ n, qk ◦ fγ � (D ∩ f−1γ [dom(qk)]) = qk ◦ fβ � (D ∩ f−1

γ [dom(qk)]),

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(3) if k + 1 ≤ n, then D ∩ f−1γ [dom(qk+1)] = D ∩ f−1

β [dom(qk)].

If D ∩ f−1γ [dom(qn)] = D ∩ f−1

β [dom(qn)], let qn+1 = qn. Otherwise, let Dn+1 = D ∩

f−1β [dom(qn)], and let qn+1 = qn ∪ {〈fγ(α), qn ◦ fβ(α)〉 : α ∈ Dn+1}. Clearly the induction

hypotheses are preserved for n+ 1.

Put qω =⋃qn. (Note in the case κ = ω, D is finite, so qω = qn for some n.) By (1) and (3),

D ∩ f−1β [dom(qω)] = D ∩ f−1

γ [dom(qω)], so call this set Dω. Let q = qω ∪ {〈fβ(α), 0〉 : α ∈

D \ Dω} ∪ {〈fγ(α), 0〉 : α ∈ D \ Dω}. This q forces that g ◦ fβ and g ◦ fγ are compatible,

again in contradiction to the assumption about p1.

Corollary 6.14. Assume κ is a regular cardinal, 2<κ = κ, and λ > κ. Then there is a

κ-closed, κ+-c.c. forcing that adds a coherent, Suslin (κ+, λ, 2)-forest.

Proof. Apply Theorems 6.8 and 6.13.

Large Suslin forests can also be obtained from combinatorial principles rather than forcing.

As reported by Jech [19] [20] [21], Laver proved in unpublished work that the existence of

Suslin (ω1, ω2, 2)-forests follows from Silver’s principle W and ♦, both of which hold in L.

Unfortunately, Laver’s proof seems to be lost to history. In trying to reconstruct it, we

encountered technical issues that led to the development of a new combinatorial principle,

which we prove consistent from a Mahlo cardinal, that can be used to construct large Suslin

forests. The main appeal for us is that, unlike the above forcing constructions, it allows a

Suslin (κ, κ+, 2)-forest to be generically added to any model with sufficiently large cardinals

using a forcing of size κ rather than κ+.

Let us establish some notation concerning trees. Suppose T is a κ-tree and α < κ. Tα is

the set of nodes at level α. If b is a cofinal branch in T , πα(b) is the node at level α in b. If

β < α, and x ∈ Tα, πα,β(x) is the node in Tβ below x.

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Definition. Wκ(λ) is the statement that there is a κ-tree T , a set of cofinal branches B,

and a sequence 〈Wα : α < κ〉 with the following properties:

(1) |B| = λ.

(2) For each α, |Wα| < κ, and Wα ⊆ P(Tα).

(3) For every z ∈ Pκ(B), there is an α < κ such that for all β ≥ α, πβ[z] ∈ Wβ.

Let T , B, 〈Wα : α < κ〉 be as above. If z ∈ Pκ(B), say “z is captured at α” when for all

β ≥ α, πβ[z] ∈ Wβ and πβ � z is injective. If z ∈ Wα and γ < α, say “z is captured at γ”

when for all β such that γ ≤ β < α, πα,β[z] ∈ Wβ and πα,β � z is injective.

Definition. W ∗κ (λ) asserts Wκ(λ), and that there exists a stationary S ⊆ κ and a sequence

〈Aα : α < κ〉 with each Aα ⊆ W 2α, such that the following additional clauses hold:

(4) κ = µ+ for a regular cardinal µ, and each Wα is a µ-complete subalgebra of P(Tα)

containing all singletons.

(5) For all α ∈ S, {z ∈ Wα : z is captured below α} is closed under arbitrary < µ sized

unions and taking subsets which are in Wα.

(6) If f : κ → Pκ(B)2 is such that |⋃α<κ f0(α) ∪ f1(α)| = κ, let 〈bα : α < κ〉 enumerate

the elements of⋃α<κ f0(α) ∪ f1(α). The set of α ∈ S with the following properties is

stationary:

(a) {bβ : β < α} is captured at α.

(b) If z ⊆ {πα(bβ) : β < α} is captured below α, then sup{β : πα(bβ) ∈ z} < α.

(c) {〈πα[f0(β)], πα[f1(β)]〉 : β < α} = Aα.

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It is easy to see that Wκ(λ) implies 2<κ = κ, and in fact Wκ(κ) is equivalent to 2<κ = κ.

If κ = µ+ and S forms part of the witness to W ∗κ (λ), then clause (4) implies µ<µ = µ, and

clause (6) can be used to show ♦κ(S). On the other hand, it follows from the next theorem

that W ∗κ (λ) prescribes no value for 2κ, besides that λ ≤ 2κ.

Theorem 6.15. Suppose κ is a Mahlo cardinal and µ < κ is regular. If G ∗H ⊆ Col(µ,<

κ) ∗ Add(κ) is generic, then V [G ∗H] satisfies W ∗κ (2κ).

Proof. In V , let T be the complete binary tree on κ, and let B be the set of all branches.

For α < κ, let Gα = G ∩ Col(µ,< α), and let Wα = P(Tα)V [Gα]. Let S = {α < κ : α is

inaccessible in V }. In V [G], fix enumerations 〈sαβ : β < µ〉 of the W 2α, and in V [G ∗H], let

Aα = {sαβ : H(α + β) = 1}. Let us check each condition.

(1) (2κ)V = (2κ)V [G∗H], so V [G ∗H] � |B| = 2κ.

(2) Since κ is inaccessible, each Wα is collapsed to µ.

(3) Suppose z ∈ Pκ(B). There is some α < κ such that z ∈ V [Gα]. For β ≥ α, πβ[z] ∈ Wβ.

(4) The regularity of µ is preserved, and clearly each Wα contains all singletons. Let 〈aξ :

ξ < δ〉 ⊆ Wα with δ < µ. Each aξ ∈ A is τGαξ for some Col(µ,< α)-name τξ. By the

µ-closure of Col(µ,< κ), 〈τξ : ξ < δ〉 ∈ V , so 〈aξ : ξ < δ〉 ∈ V [Gα].

(5) By the Mahlo property, S is stationary, and by the κ-c.c. of Col(µ,< κ) and κ-closure

of Add(κ), it remains stationary in V [G ∗H]. Suppose α ∈ S.

(a) Unions: Let A ∈ Pµ(Wα) have the property that all a in A are captured below

α. As above, A ∈ V [Gα]. Now in V [Gα], α = µ+ and |Tβ| = µ for β < α. So

if πα,β � a is injective, then V [Gα] � |a| < α, and thus V [Gα] � |⋃A| < α. For

distinct x, y ∈⋃A, let γx,y < α be the least γ such that πα,γ(x) 6= πα,γ(y). We have

γ = sup{γx,y : x, y ∈⋃A} < α. Hence if γ ≤ β < α and all a ∈ A are captured at

β, then⋃A is captured at β.

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(b) Subsets: Suppose z0 ∈ Wα is captured below α, and z1 ∈ Wα is a subset of z0.

Then V [Gα] � |z1| < α, so by the α-c.c. of Col(µ,< α), there is some β < α such

that z1 ∈ V [Gβ]. Thus z1 is captured below α.

(6) First work in V [G]. Let f be an Add(κ)-name for a function from κ to Pκ(B)2, and

let 〈bα : α < κ〉 be as in clause (6). Let C be a name for a club, and let p0 ∈ Add(κ)

be arbitrary. Build a continuous decreasing chain of conditions below p0, 〈pα : α <

κ〉 ⊆ Add(κ), and a continuous increasing chain of ordinals, 〈ξα : α < κ〉 ⊆ κ, with the

following properties: For all α,

• pα+1 ξα ∈ C,

• pα+1 decides f � dom(pα) and {bβ : β < α},

• dom(pα+1) is an ordinal > max{dom(pα), ξα, α}, and

• ξα+1 > dom(pα+1).

Let g : κ → Pκ(B)2 and {bα : α < κ} be the objects defined by what the chain

〈pα : α < κ〉 decides. For each α < κ, there is a predense set Eα ⊆ Col(µ,< κ) of size

< κ such that g(α) and bα are decided by elements of Eα. There is a club D ∈ V such

that ∀α ∈ D, ∀β < α, Eβ ⊆ Col(µ,< α). For α ∈ D, g � α and {bβ : β < α} are in

V [Gα].

Back in V [G], for α < κ, let γα be the least γ ≥ α such that πγα � {bβ : β < α}

is injective. If α is closed under β 7→ γβ, then γα = α. As S is stationary, there is

α ∈ S ∩ D such that γα = α, ξα = α, and pα α ∈ C. We have that {bβ : β < α} is

captured at α, and that {〈πα[g0(β)], πα[g1(β)]〉 : β < α} ⊆ W 2α. Since α is inaccessible in

V , if z ⊆ {πα(bβ) : β < α} is captured below α, then V [Gα] � |z| < α, so {β : πα(bβ) ∈ z}

is bounded below α.

Let q ≤ pα be such that for β < µ, q(α + β) = 1 if sβα = 〈πα[g0(β)], πα[g1(β)]〉, and

q(α + β) = 0 otherwise. Then q α ∈ C ∩ S, and that items (a), (b), and (c) in clause

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(6) hold at α. As p0 was arbitrary, clause (6) is forced.

Question. Can W ∗κ (λ) be forced without the use of large cardinals? Can it be forced in a

cardinal-preserving way? Does L � “For all regular κ, W ∗κ+(κ++)”?

Theorem 6.16. W ∗κ (λ) implies there is a coherent, Suslin (κ, λ, 2)-forest.

Proof. Let κ = µ+, and let T , B, 〈Wα : α < κ〉, 〈Aα : α < κ〉, and S ⊆ κ witness W ∗κ (λ).

We will construct a sequence of functions 〈fα : α < κ〉 on the nodes of T that will generate

a coherent family of functions on B with the desired properties. Each fα will have domain

Tα and range contained in {0, 1}.

Let f0 be a function from T0 to 2. Assume we have have constructed a sequence of functions

〈fβ : β < α〉, with each fβ : Tβ → 2, satisfying the following property:

(∗) If r ∈ Wβ is captured at γ < β, then fβ � r disagrees with fγ ◦ πβ,γ � r on a set of size

< µ.

Let Rα = {r ∈ Wα : r is caputured below α}. Consider the set Fα of partial functions on

Tα of the form fγ ◦ πα,γ � r for r ∈ Rα and γ witnessing its membership in Rα. Assume

γ0 < γ1 and fγ0 ◦ πα,γ0 � r0 and fγ1 ◦ πα,γ1 � r1 are in Fα. By hypothesis (∗), fγ1 disagrees

with fγ0 ◦πγ1,γ0 at less than µ many points in πα,γ1 [r0]. Therefore, there are less than µ many

points in r0 ∩ r1 at which fγ0 ◦ πα,γ0 and fγ1 ◦ πα,γ1 disagree. So Fα is a µ-coherent family.

Assume first that α /∈ S. Using Lemma 6.3(a), let fα : Tα → 2 be such that {fα} ∪ Fα is

µ-coherent. Then (∗) holds for 〈fβ : β ≤ α〉.

Now assume α ∈ S. Let Hα be the closure of Fα under < µ modifications. Consider Hα as

a partial order with f ≤ g iff f ⊇ g. The set Aα ⊆ W 2α codes a set of relations from subsets

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of Tα to 2. If 〈a0, a1〉 ∈ Aα, construct a relation h by putting 〈x, i〉 ∈ h iff x ∈ ai, and call

the set of all such things A′α. It may be the case that every member of A′α is a function

and a member of Hα, and that A′α is a maximal antichain in Hα. If not, ignore all these

considerations, and let fα be as in the case α /∈ S, so that (∗) is preserved.

Suppose A′α is a maximal antichain in Hα. Enumerate Rα as 〈rβ : β < µ〉. By clauses (4)

and (5) of the definition of W ∗, Rα is closed under unions of size < µ. Hα is also a µ-closed

partial order. If 〈hi : i < β < µ〉 is a decreasing sequence, then⋃i<β dom(hi) = r ∈ Rα, so

let γ witness this. By (∗), each hi disagrees with fγ ◦πα,γ on a set of size < µ, and so⋃i<β hi

does as well by the regularity of µ.

Setting sβ =⋃ξ<β rξ, we have 〈sβ : β < µ〉 is an increasing cofinal sequence in Rα. For

β < µ, let γβ be the least γ < α that witnesses sβ ∈ Rα. Let 〈tβ : β < µ〉 enumerate

all < µ sized subsets of Tα, such that each subset is repeated µ many times. For a partial

function f : Tα → 2 and β < µ, let f/tβ be f with its output values switched at the points

in dom(f) ∩ tβ.

We will define fα inductively as⋃β<µ hβ. Let h0 = ∅. Assume 〈hi : i < β〉 has been chosen

so that:

(1) for i < j < β, hi ⊆ hj;

(2) for i < β, dom(hi) = sξi where ξi ≥ i, and ξi > ξj for j < i;

(3) for i < β, there is a ∈ A′α such that hi+1/ti is a common extension of hi/ti and a.

Given hi, there is some a ∈ A′α that is compatible with hi/ti. Let ξi+1 > ξi be such that

sξi+1⊇ dom(a) ∪ sξi , and let g ∈ Hα be a common extension of a and hi/ti with domain

sξi+1. Let hi+1 = g/ti. Clearly (1)–(3) are preserved at successor steps. At limit steps β,

we set hβ =⋃i<β hi. This is in Hα as well by µ-closure, and the preservation of (1)–(3) is

trivial.

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The point is this: For every t ∈ Pµ(Tα), fα/t extends some a ∈ A′α. For let i < µ be

large enough that sξi ⊇ t and ti = t. Then by (3), hi+1/t extends some a ∈ A′α, and

hi+1/t = (fα/t) � sξi+1. We also check that (∗) is preserved at α: Every r ∈ Rα is covered

by some sξi , and fα � sξi = hi, which coheres with fγ ◦ πα,γ � sξi when sξi is captured at γ.

Now we define the forest. For z ∈ Pκ(B), let γz be the least γ < κ such that z is captured

at γ. Let fz : z → 2 be fγz ◦ πγz � z. Let F be the closure of {fz : z ∈ Pκ(B)} under < µ

modifications. Note that by (∗), if β ≥ γz, then fβ ◦ πβ � z disagrees with fz at < µ many

points. Hence F is a coherent (κ,B, 2)-forest.

Finally, we verify the κ-c.c. First note that F satisfies the κ+-c.c. by a delta-system argu-

ment. So assume towards a contradiction that A = {aα : α < κ} is a maximal antichain.

Let zα = dom(aα), and code each aα as 〈z0α, z

1α〉, where ziα = {b : aα(b) = i}. Let 〈bα : α < κ〉

enumerate the elements of⋃α<κ zα. Define:

• C0 = {α < κ :⋃β<α zβ = {bβ : β < α}}.

• C1 = {α < κ : {aβ : β < α} is a maximal antichain contained in {f ∈ F : (∃η <

α) dom(f) ⊆ {bβ : β < η}}}.

• C2 = {α < κ : (∀β < α)γz0β , γz1β , γzβ < α}.

It is easy to see that C0, C1, and C2 are club. By clause (6) of the definition of W ∗, let

α ∈ S ∩C0∩C1∩C2 be such that {bβ : β < α} =⋃β<α zβ is captured at α, all z ⊆ {πα(bβ) :

β < α} captured below α have sup{β : πα(bβ) ∈ z} < α, and Aα = {〈πα[z0β], πα[z1

β]〉 : β < α}.

We claim A′α is a maximal antichain in Hα. For β < α, zβ is captured below α since α ∈ C2,

so the function coded by 〈πα[z0β], πα[z1

β]〉 is in Hα. If h ∈ Hα is incompatible with every

member of A′α, then consider z = {bβ : β < α and πα(bβ) ∈ dom(h)}, and let f = h ◦ πα � z.

Clauses (4) and (5) imply πα[z] is captured below α, so sup{β : bβ ∈ z} < α. Since

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α ∈ C1, f is compatible with some aβ with β < α. But aβ is coded and projected down as

〈πα[z0β], πα[z1

β]〉 ∈ Aα, so h is compatible with some member of A′α after all.

Since {bβ : β < α} is captured at α, the construction has sealed this antichain. Consider

any other f ∈ F such that dom(f) ⊇ {bβ : β < α}. Then f � {bβ : β < α} is a < µ

modification of fα ◦ πα � {bβ : β < α}. By the above argument, all < µ modifications of fα

extend a member of A′α, and so f is compatible with some aβ, β < α. This contradicts the

assumption that A = {aγ : γ < κ} is an antichain.

Finally, we may answer the question of whether ZFC+GCH proves an analogue of Taylor’s

theorem above ω1. Start with an almost-huge cardinal κ and a Mahlo cardinal µ < κ.

Suppose κ carries a tower of height δ, and λ is regular such that κ ≤ λ. By Theorem 2.17, if

X ∗H is A(µ, κ) ∗ Col(λ,<δ)-generic, then in V [X][H] there is a normal, fine, κ-complete,

λ-dense ideal on Pκ(λ). Furthermore, this forcing is µ-strategically closed, and it is easy to

show that µ-strategically closed forcings preserve stationary subsets of µ. Thus µ remains a

Mahlo cardinal in V [X][H]. If ν < µ is regular and G is Col(ν,< µ)∗Add(µ)-generic, then in

V [X][H][G] there is a coherent (µ, κ, 2)-Suslin forest, and thus a µ-Suslin algebra of uniform

density κ. Since Col(ν,< µ) ∗ Add(µ) is µ-dense, it preserves the density of κ-complete

ideals.

Since the µ-Suslin algebra is µ-distributive forcing with it preserves the equation 2ν = µ. If

we force with this Suslin algebra over V [X][H][G], then Lemmas 5.9 and 5.10 imply that in

the generic extension, there is a normal, fine, κ-complete, nonregular ideal on Pκ(λ), but no

λ-dense ideals. Hence we have the following consistency result:

Theorem 6.17. If ZFC+“There is an almost-huge cardinal” is consistent, then for m ≥

n ≥ 2, ZFC+GCH does not prove the statement, “If there is a nonregular ideal on Pωn(ωm),

then there is a dense ideal on Pωn(ωm).”

99

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