1.4. Why strictly first-order hypotheses are not enough . . . . . . . . . 5
2. Outer model theories . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1. Weak outer models . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. Strong outer models . . . . . . . . . . . . . . . . . . . . . . . 9
2.3. Outer model logic . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4. The complexity of somth(V, ∅)+ . . . . . . . . . . . . . . . . . . 12
3. Global definability of theories satisfiable in outer models . . . . . . . . 14
3.1. Well-founded direct limits . . . . . . . . . . . . . . . . . . . . 15
3.2. Proof of the global version of the main theorem . . . . . . . . . . . 16
4. Hyp-sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3. Hyp-sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5. Local definability of theories satisfiable in outer models . . . . . . . . . 26
5.1. Richness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1. Introduction
This paper solves the anticharacterization problem first identified in [S1]. Perhaps the most striking instance of this problem is that, working relative to just ZFC, there is no uniform first-order definition of the family of subsets of ω2 that have a closed unbounded (club) subset in some ω1 and ω2 preserving outer model. Other examples concern large homogeneous sets for partitions and cofinal branches through trees, as well as the club susbet problem at successors of singular cardinals, cf. [S2] and [S3]. The same phenominon has been considered from a different perspective by S. Fried- man, cf. Theorem 7.28 in [F].
A consequence of the main theorem of this paper is that over a sufficiently non- minimal model of ZFC + “Ramsey cardinals are definably stationary,” all of these characterization problems are solvable.
Key words and phrases. outer model, combinatorial characterization, hyp-sharp, stationary set
Research supported by N.S.F. Grant DMS 0501114.
1. INTRODUCTION
“Outer models” and “sufficient non-minimality,” both of which come in degrees, are discussed later in this section.
A subset of ω1 has a club subset in an ω1-preserving outer model exactly when it is stationary—a property that is Π1 in the parameter ω1 over Hω2 . The characterization we get for subsets of ω2 is Π1 in the parameter ω2 over Hω3 .
Anti-characterization results are the motivation for this paper, but this paper is not directly about the “club subset problem”, or any other characterization problem, for that matter. We use elementary model theory and “hyp-sharps”, a generalized version of ordinary sharps of sets, to prove a general theorem. The proof does not require any high-powered machinery. Indeed, it would be accessible to a reader who has been asleep for the past twenty years; the motivation is modern.
In order to state the main theorem precisely, we need some definitions.
1.1. Outer models
If V is a standard transitive model of ZFC, say that W ⊇ V is a weak outer model of V if W is also a standard transitive model of ZFC and W ∩ OR = V ∩ OR. If, in addition, (W ;V ) satisfies ZFC in a language with a predicate symbol for the inner model V , let us say the W is a strong outer model of V .
Note that any model is a strong outer model of itself. The reader is readily excused if he has not contemplated the distinction between
strong and weak outer models before. Outer models constructed by forcing are strong. The ambient universe is a strong outer model of any of its definable inner models. On the other hand, weak outer models are perhaps more relevant to problems such as the club subset problem at ω2. Given an X ⊆ ω2, the question is whether there is an obstacle to
∃C( C ⊆ X ∧ “C is club in ω2” ∧ ω1 = ωV
1 ∧ ω2 = ωV 2
)
being satisfiable. This formula has set parameters—constant symbols for X, ωV 1 , and
ωV 2 —but does not involve a predicate symbol for V as an inner model of some outer
model. Quantification over outer models is inherently second-order. Nevertheless, we have
no interest in working in some second-order set theory. Specifying such a theory, or substituting some specific notion of class forcing for talk of general outer models, risks begging the questions that motivate this investigation.
The least restrictive way to understand talk of general outer models of V is to understand “V ” as a countable standard set model in some larger ambient universe. The use of the letter ‘V ’ is intended to suggest just that this model is proxy for the real universe, which in turn is “real” at least in the sense that our entire discussion is formalizable in first-order ZFC. This use of the letter ‘V ’ is intended to indicate the distinguished role of that model, as compared to that of its outer and inner models. Typically we use ‘W ’ for outer models of V and ‘M ’ for definable inner models. When the context makes it unambiguous, we write ‘∞’ for V ∩OR.
It is not difficult to construct a pair of models W and V such that W is a weak, but not a strong outer model of V . With a somewhat stronger non-minimality hypothesis than is required for the rest of our work, any statement in the language of set theory
2
1.1. OUTER MODELS
that is satisfiable in some weak outer model is also satisfiable in some strong outer model. It is open whether this extra non-minimality is necessary, cf., Question 3.8.
1.2. Sufficient non-minimality A minimal model is a standard transitive model of ZFC in which there exists a set x such that every element of its universe is first-order definable from parameters in x.
“Sufficiently non-minimal” is intended to be like “large cardinal axiom”. Both indicate a calibrated range of statements of a specific sort. The analogy is better than that. In this paper, a model V is sufficiently non-minimal when ∞ has one of several large cardinal properties in Hyp(V ). Hyp(V ) is the smallest admissible set with V as an element; that is, Hyp(V ) = Lα(V ), where α is least such that this structure satisfies Σ0 separation and collection.
For the most part, it will be enough to know that∞ is definably regular in Hyp(V )— in our analogy, that ∞ is inaccessible, or at least inaccessible in Hyp(V ). Definably regular just means that in Hyp(V ) there is no definable function from a smaller ordinal that is cofinal in ∞ = V ∩ OR. At a few points we need more, namely, that Hyp(V ) satisfies that measurable cardinals or Ramsey cardinals are stationary in ∞, or that “hyp-sharps” are “definably rich”.
Later in this section we show that strictly first-order hypotheses do not suffice for any theorem like the main theorem to hold.
1.3. The axioms ZFC+
One hypothesis of the main theorem of this paper is that V satisfies
ZFC+ = ZFC + “hyp-sharps are definably rich”. (1.1)
A hyp-sharp is an analog of the ordinary sharp of a set, given by indiscernibles for a structure of the form
(Hyp(Vκ);Vκ, <, a)a∈r,
where κ is a Ramsey cardinal, < is a well-ordering of Vκ, and r ∈ Vκ is a transitive set. Section 4 develops hyp-sharps from scratch.
This machinery is used for two other purposes in forthcoming work. One is to investigate limitations on the ability of set forcing to witness the satifiability of first- order statements in outer models, assuming that both V and those outer models satisfy arbitrary large cardinal hypotheses. The other concerns a variant of S. Friedman’s Inner Model Hypothesis (cf. [F2] and [FWW]) in which both V and its outer models are required to contain arbitrary large cardinals. Models of Friedman’s IMH contain no large cardinals.
A class R ⊆ V is definably rich if, for each a ∈ V and each n ∈ ω, there exists x ∈ V such that
a ⊆ x Σn V
and tc(x), the transitive collapse of x, is an element of R. It is easy to check, cf. Proposition 5.2, that
definably stationary ⇒ definably rich ⇒ unbounded
3
1.3. THE AXIOMS ZFC+
and no converses hold. Hyp-sharps, like ordinary sharps, are blueprints for Ehrenfeucht-Mostowski models.
“Hyp-sharps are definably rich” is shorthand for the statement that the class of all Ehrenfeucht-Mostowski models generated from hyp-sharps is definably rich.
For the sake of giving a precise statement of the main theorem without first going through the development of hyp-sharps, let us temporarily set
ZFC+ = ZFC + “Ramsey cardinals are definably stationary”. (1.2)
This just means that every definable closed unbounded class of ordinals contains a Ramsey cardinal. It follows from Lemma 4.6 that the theory on line (1.2) implies the theory on line (1.1).
Main Theorem 5.1. There exists a formula good(x) in the language of set theory with the following properties.
Let V be a standard transitive model of ZFC+. Assume that κ is a regular un- countable cardinal in V and that T ∈ HV
κ is a set of axioms in the language of set theory with parameters in HV
κ .
(1) If HV κ ² good[T ], then there exists a weak outer model of V that satisfies T .
If also V is sufficiently non-minimal, then there exists a strong outer model of V that satisfies T .
(2) If HV κ ² ¬good[T ] and V is sufficiently non-minimal, then T is not satisfied in
any weak (a fortiori, strong) outer model of V .
The formula good(x) can be taken to be parameter-free Π2. If κ > ω1 and r ∈ Hκ
is uncountable, then good(x) can be taken to be Π1 in the parameter r.
More non-minimality is required by the proof for the second sentence in (1) than is for (2). Part (2) requires that ∞ is definably regular in Hyp(V ). For a strong outer model, in part (1) it enough if Hyp(V ) satisfies that Ramsey cardinals are definably stationary in ∞.
Most of our work for Theorem 5.1 will concern weak outer models. We first prove all but the second sentence in (1). For it, we show that its extra non-minimality as- sumption implies that if T is satisfiable in some weak outer model, then T is satisfiable in some strong outer model.
If we delete the second sentence in conclusion (1), then we have a theorem simply about satisfiability in weak outer models. With only first-order hypotheses, if T is good, then T is satisfiable in a weak outer model. Otherwise, there exists a set witness that T is not satisfiable in any weak outer model. That set witness is only reliable if V is sufficiently non-minimal. How can this be? The witness is, more or less, an iterable set model of the form Hyp(N) ∈ V . Iterating through all of the ordinals of V , the model N stretches to a definable inner model. The witness is reliable only if Hyp(N) stretches to a well-founded structure. This will be the case if ∞ is definably regular in Hyp(V ).
A natural question is whether it is possible to strengthen the conclusions in (1) so that if V is sufficiently non-minimal and T is good, then T is satisfiable in a sufficiently non-minimal outer model of V . The proof of Proposition 1.1 below shows that this is not possible.
4
OUTER MODEL SATISFIABILITY
1.4. Why strictly first-order hypotheses are not enough Talk of satisfiability in outer models is second-order. The point of the main theorem is to reduce it to a first-order property good in models of ZFC+. But the non-minimality hypotheses in Theorem 5.1 are not first-order. What progress is made by replacing one second-order statement with another?
The purpose of this subsection is to observe that strictly first-order hypotheses do not suffice for a theorem like the main theorem, then to argue that the non-first-order hypothesis we need is reasonable.
An easy application of the fixed-point lemma shows that something more than a recursive extension of ZFC is needed to get any theorem along the lines of Theorem 5.1. The argument works equally well for strong and weak outer models, so we write simply “outer model” for either.
Proposition 1.1. Assume that
• ZFC? is a recursive first-order theory extending ZFC and that ZFC? has count- able standard transitive models;
• good(x) is a parameter-free formula such that if V is a countable standard transitive model of ZFC?, T ∈ V is set of axioms and V ² good[T ], then V has an outer model that satisfies T ; and
• bad(x) is a parameter-free formula such that if V is a standard transitive model of ZFC?, T ∈ V is set of axioms, and V ² bad[T ], then V does not have an outer model that satisfies T .
Then there exists a recursive T ⊇ ZFC? such that neither good[T ] nor bad[T ] holds in any countable standard transitive model of ZFC?.
Proof: Let be a sentence in the language of set theory such that
ZFC ` ↔ bad(ZFC∗ + ).
Let T = ZFC? + , and let V be a countable standard transitive model of ZFC?. On the one hand, if V ² bad[T ], then V ² . But then V is an outer model of itself
satisfying T = ZFC? + . So V ² ¬bad[T ]. On the other hand, if V ² good[T ], then V has an outer model W that satisfies
ZFC? + . But this is impossible, because then W ² bad[T ] even though W is an outer model of itself satisfying ZFC? + .
So V ² ¬good[T ].
The reader may wonder what happens if this argument is run against the formula good of the main theorem with ¬good in the role of bad and ZFC+ in the role of ZFC?. In this case V ² good[T ]. If W is an outer model satisfying ZFC+ +, then W is not sufficiently non-minimal. Indeed, any hyp-sharp in W that witnesses ¬good[ZFC++] is an explicit witness that W is not sufficiently non-minimal. That is, it is first-order in W that W satisfies any fragment of ZFC+ + of bounded complexity, and it is first-order in W that there exists a hyp-sharp witnessing that T is ¬good. Stretching any such hyp-sharp through the ordinals of W yields a non-well-founded direct limit. To the extent that W “knows” that it is a model of ZFC+ + , it “knows” that it is not sufficiently non-minimal.
5
1.4. WHY STRICTLY FIRST-ORDER HYPOTHESES ARE NOT ENOUGH
What is the point of proving anything like the main theorem of this paper? A special Aronszajn tree cannot have a cofinal branch in any ω1-preserving outer
model for a concrete combinatorial reason. A Souslin tree can have a cofinal branch in an ω1-preserving outer model, again for a good reason.
A “good reason” is simply a witness of some sort that a given statement is or is not satisfiable in some weak outer model. For this purpose, weak outer models are perhaps the right notion because we are we are considering the satisfiability of statements that do not contain a predicate symbol for V . (The project is impossible otherwise, in view of Proposition 2.9.)
It is incompatible with ZFC that everything that potentially could exist actually does exist—there are no existentially complete models of ZFC. Generalized Martin’s Axioms and corresponding generic absoluteness statements imply limited degrees of existential completeness, enforcing that the universe is rich. The same argument is often presented regarding large cardinal hypotheses. A case can be made that “richness” is in accord, or even implicit in the intuition behind the definition of the cumulative hierarchy of pure sets.
Beyond a fairly low level of logical complexity, existential completeness is simply impossible. Next best is to ask which existential statements are potentially satisfiable and which are simply impossible. With this understanding, the anti-characterization phenomenon indicates that the universe of sets is fundamentally irrational. In the example with which we began, there is no uniform way of separating the subsets of ω2 that potentially have a club subset without collapsing ω1 or ω2 from those that do not.
Irrationality might simply be a fact. But maybe not. The theorem is that, for every statement in the language of set theory with parameters in V , there is a good reason for that statement to be potentially satisfiable or there is a good reason for that statement to be simply unsatisfiable, provided that V has two properties: Large cardinals are plentiful in V , and ∞ itself satisfies a mild (but second-order) large cardinal hypothesis. Assuming just the first-order part of this (that large cardinals are plentiful), to suppose that good fails to characterize the statements that are satisfiable in a weak outer model is to suppose that the universe as a whole resembles Vκ, where κ is a singular strong limit cardinal in L(Vκ). To the author’s mind, this is at odds with the intuition supporting the Collection Axiom.
2. Outer model theories The proof of the main theorem is divided among the remaining sections of this paper. In this and the next section we prove a version of the main theorems that does not require hyp-sharps, using instead measurable cardinals, which are more familiar. The cost is that we only get a global definition of the set of theories that are satisfiable some outer model. This section develops the elementary model theory that is used in both section 3 for the global version of the main theorem and in section 5 for the real McCoy.
In section 4 we develop hyp-sharps and in section 5 we prove the main theorem with hyp-sharps in the role taken in section 3 by iterated ultrapowers.
This section has four subsections. The first concerns weak outer models. The second subsection proves analogous results for strong outer models. The third subsection
6
2. OUTER MODEL THEORIES
extends the results in the first two through the point at which the weak and strong outer model theories diverge. The final subsection is independent of and unnecessary for the rest of the paper. It shows that an oracle for strong outer model satisfiability in a language with a predicate symbol for V as an inner model is, more or less, a master code for Hyp(V ), hence cannot be definable over V .
2.1. Weak outer models The language of set theory has the single two-place relation symbol “∈” (together with “=” for equality, of course). If M is a set or class, the language of set theory with parameters in M augments the language of set theory with a constant symbol a corresponding to each element a ∈M .
The weak outer model language. Let M be a transitive set or class satisfying ZFC. The language LM extends the language of set theory with parameters in M by adding a one-place function symbol F :
LM = {∈, F } ∪ { a : a ∈M }. The function symbol F serves to form closed terms to name the elements of an outer model.
The M-suitable formulas are defined as follows. The quantifier-free M -suitable formulas are the infinitary Boolean combinations of
atomic formulas of LM that lie in M and in which only finitely many distinct variables occur:
• Any atomic formula of LM is a quantifier-free M -suitable formula. • If Φ(x1, . . . , xk) ∈ M is a set of quantifier-free M -suitable formulas in which
only the variables x1, . . . , xk occur, then ∧
Φ is an M -suitable formula. • If is an M -suitable formula, then so is (¬).
General M -suitable formulas are the closure of the quantifier-free M -suitable for- mulas under finitary quantification and finite Boolean combinations:
• Any quantifier-free M -suitable formula is an M -suitable formula. • If and ψ are M -suitable formulas, then so are (¬) and ( ∧ ψ). • If is an M -suitable formula and x is a variable, then ∀x is an M -suitable
formula.
The weak outer model theory womth(M, T ). Let T ∈ M be a set of first-order sentences in the language of set theory with parameters in M . Let womth(M,T )—the weak outer model theory of T over M—be the following set of M -suitable axioms in LM :
T
∀x(x ∈ a↔ ∨ b∈a x = b), for each a ∈M
∀y∃x(x ∈ OR ∧ y = Fx)
∀x(x /∈ OR → Fx = ∅) ∀x∀z(z ∈ Fx→ ∃y ∈ x(z = Fy))
7
2.1. WEAK OUTER MODELS
Finally, let ΘM∩OR(x) = {x ∈ OR} ∪ {x 6= β : β ∈M ∩OR }.
Lemma 2.1. Let V be a countable standard transitive model of ZFC. Let T ∈ V be a set of axioms in the language of set theory, perhaps with parameters in V . The following are equivalent:
(1) V has a weak outer model that satisfies T .
(2) womth(V, T ) has a model that omits ΘV ∩OR.
The proof is elementary. The only reason to give details is to observe that T contains all of the necessary axioms.
Proof of (1) ⇒ (2): Let W be a weak outer model of V that satisfies T . We expand (W,a)a∈V with a function F :W →W satisfying the last three axioms of womth(V, T ). For this it suffices to introduce an W -amenable function G with domain ∞ = OR∩V and range W satisfying the last axiom of womth(V, T ).
We do this by class forcing. Work in (W ; a)a∈V . Conditions are functions p:α→W such that, p(γ) ⊆ p”γ, for all γ ∈ α, ordered by reverse functional extension.
Note that in (W ; a)a∈V these conditions are closed under unions of chains of length less than ∞.
Given any β and any condition p, it is trivial to extend p to a condition q including β in its domain, since we can set q(γ) = ∅, for γ /∈ dom(p).
To see that { p : a ∈ rng(p) } is dense, for a ∈ W , proceed by ∈-induction on W . Suppose that p is a condition. By induction (using Choice and <∞-closure), we may assume that a ⊆ rng(p). Then
q = p ∪ {(dom(p), a)}
is as required. This completes the proof of (1) ⇒ (2).
Proof of (2) ⇒ (1): Suppose that A = (W ;∈A, FA, aA)a∈V satisfies womth(V, T ) and omits ΘV ∩OR. Because womth(V, T ) includes ∀x(x ∈ a↔ ∨
b∈a x = b) we may assume that V ⊆ W , that aA = a, for a ∈ V , and that ∈A restricted to V is the standard ∈-relation. It also follows that { aA : a ∈ V } is transitive under ∈A. Because A omits ΘV ∩OR, it follows that ORA = V ∩OR.
Now W = FA”(OR ∩ V ). It follows from the last axiom that ∈A is well-founded, since OR ∩ V is. Because { aA : a ∈ V } is transitive under ∈A, the canonical isomor- phism (W ;∈A) ∼= (W ′,∈), where W ′ is transitive, fixes every element of V . So we may as well assume that W itself is a transitive set and ∈A is the standard ∈-relation on W .
Thus W ⊇ V is a standard transitive model of ZFC + T such that W ∩ OR = V ∩OR.
The operation ΓM∩OR. Our next move is to improve Lemma 2.1’s characterization of outer model satisfiability from omitting types to simple consistency. It comes at the apparent price of making the class of sentences in question undefinable over V .
8
2.1. WEAK OUTER MODELS
Fix a transitive M and define the following operation ΓM∩OR on X ⊆M consisting of LM -sentences:
∈ ΓM∩OR(X) iff is a sentence of LM and either X ` or = ∀x(x ∈ OR → ψ(x)),
where X ` ψ(α), for every α ∈M ∩OR
If X ⊆ M is a class of sentences of LM , let X+ be the least fixed point of ΓM∩OR
containing X, that is, the smallest Y ⊇ X such that ΓM∩OR(Y ) = Y . Then • X ⊆ X+. • If X is definable over M , then X+ is Σ1 definable over Hyp(M). In particular,
if T ∈M is a set of sentences, then womth(M,T )+ is Σ1 definable over Hyp(M) uniformly in the parameters T and M .
• X+ locally omits ΘM∩OR—if X+ + ∃x(x ∈ OR ∧ ψ) is consistent, then there exists α ∈ OR ∩M such that X+ + ψ(α) is consistent. Indeed,
• X+ is the smallest deductively closed set extending X and locally omitting ΘM∩OR. (Observe that if Y is deductively closed and locally omits ΘM∩OR, then Y is a fixed point of ΓM∩OR.)
Lemma 2.2. Let V be a countable standard transitive model of ZFC and let T ∈ V be a set of sentences of the language of set theory, perhaps with parameters in V . The following are equivalent:
(1) T is satisfied in some weak outer model of V .
(2) womth(V, T )+ is consistent.
Proof of (1) ⇒ (2): By the Lemma 2.1, womth(V, T ) has a model omitting ΘV ∩OR. Let Y be the set of sentences of LV satisfied in this model. Then Y is a fixed point of ΓV ∩OR. Consequently, womth(V, T )+ ⊆ Y .
Proof of (2) ⇒ (1): Because womth(V, T )+ is countable, consistent, and locally omits ΘV ∩OR, it has a model omitting ΘV ∩OR. By the Lemma 2.1, V has a weak outer model satisfying T .
The final step in the proof of the main theorem is to find in V a first-order equivalent of “womth(V, T )+ is consistent”. That is where large cardinals will come in. Before going on to that, we need some results about strong outer models.
2.2. Strong outer models
LM (V ) = LM ∪ {V } = {∈, F , V } ∪ { a : a ∈M },
where V is a one-place predicate symbol for M as an inner model of some outer model. The M -suitable formulas of this larger language are defined just as before.
9
2.2. STRONG OUTER MODELS
The strong outer model theory somth(M, T ). Let T be a set of first-order axioms in the language of set theory, perhaps employing parameters in M . Beware that we do not allow “ V ” to occur in T . Proposition 2.9 explains the necessity of this restriction.
Let somth(M,T )—the strong outer model theory of T over M—be the following set of axioms in LM (V ):
womth(M,T )
ZFC in the augmented language {∈, V } ∀x(V x ∧ rk(x) < α↔ x ∈ Vα), for each α ∈M ∩OR
Lemma 2.3. Let V be a countable standard transitive model of ZFC. Let T ∈ V be a set of axioms in the language of set theory, perhaps employing parameters from V . The following are equivalent:
(1) V has a strong outer model that satisfies T .
(2) somth(V, T ) has a model that omits ΘV ∩OR.
Proof: (1) ⇒ (2) is the same as for Lemma 2.1. For (2) ⇒ (1), suppose that A = (W ;V A, FA, aA)a∈V satisfies somth(V, T ) and omits ΘV ∩OR. Using the proof of Lemma 2.1, we may assume that W is a weak outer model of V and that aA = a, for all a ∈ V . Using the last axiom of somth(V, T ), we have that V A ∩Wα = Vα , for all α ∈ V ∩OR. But V ∩OR = W ∩OR, so V A = V . Because ZFC in {∈, V } and T are included in somth(V, T ), we conclude that W is a strong outer model of V .
The operation ΓM∩OR. This is defined exactly as for the weak outer model language, working in the larger language that includes V , of course. The same remarks enumer- ated in the previous subsection hold regarding the smallest fixed point X+ extending a given set X of sentences in the strong outer model language.
Using Lemma 2.3 in place of Lemma 2.1 as in the proof of Lemma 2.2, we have
Lemma 2.4. Let V be a countable standard transitive model of ZFC and let T ∈ V be a set of sentences of the language of set theory with parameters in V . The following are equivalent:
(1) T is satisfied in some strong outer model of V .
(2) somth(V, T )+ is consistent.
2.3. Outer model logic Let T ∈ V be a set of axioms in the language of set theory, perhaps with parameters in V . Let be a sentence in this language. Declare that
T ²som
if W ² whenever W is a strong outer model of V that satisfies T . Say that
T ²wom
if W ² whenever W is a weak outer model of V that satisfies T .
10
Lemma 2.5.
(1) T ²som iff somth(V, T )+ ` (2) T ²wom iff womth(V, T )+ `
Proof: We argue for (1); the proof of (2) is the same. We must see that
T + holds in some strong outer model of V iff somth(V, T )+ + is consistent.
By Lemma 2.4, this is equivalent to showing
somth(V, T + )+ is consistent iff somth(V, T )+ + is consistent.
For this, it suffices to see
somth(V, T + )+ = {ψ : somth(V, T )+ + ` ψ }.
The second set is included in the first because somth(V, T )+∪{} ⊆ somth(V, T +)+
and somth(V, T + )+ is deductively closed. To see that the first set in included in the second, noting that somth(V, T )∪{} =
somth(V, T + ), it is enough to show that the second set is a fixed point of ΓV ∩OR. Assume that
somth(V, T )+ + ` ψ(α), for all α ∈ V ∩OR.
Then somth(V, T )+ ` → ψ(α), for all α ∈ V ∩OR.
So somth(V, T )+ ` ∀x ∈ OR (→ ψ(x)).
Hence somth(V, T )+ + ` ∀x ∈ OR ψ(x).
The final lemma of this subsection does not generalize to strong outer model theo- ries.
Lemma 2.6. Suppose that V is a countable standard transitive model of ZFC, that M is a definable inner model of V , and that T ∈M is a set of axioms in the language of set theory, perhaps with parameters in M . Then
womth(M,T )+ ⊆ womth(V, T )+.
Proof: Suppose that ∈ womth(M,T )+. Let W be a weak outer model of V that satisfies T . Then W is also a weak outer model of M , hence W ² . Since W is arbitrary, it follows from Lemma 2.5 that ∈ womth(V, T )+.
11
2.4. The complexity of somth(V, ∅)+
This subsection is independent of the rest of the paper. Its purpose is to explain why weak and strong outer models must be treated differently. The reader primarily interested in the main theorem can skip it.
By Lemma 2.5 we have that if somth(V, T )+ is consistent, then
V ∈ somth(V, T )+ iff V ² , (2.7)
for sentences in the language of set theory with parameters in V . Even when T = ∅, the somth(V, T )+ is more complicated than this suggests. It is almost a Σ1 master code for Hyp(V ). (See Proposition 2.10 for an exact statement.)
Just the simple observation of line (2.7) has the following two important conse- quences.
Proposition 2.8. somth(V, T )+ is definable over V iff it is inconsistent.
In contrast, we shall see in section 3 that if V is sufficiently non-minimal and measurable cardinals are definably stationary, then womth(V, T )+ is definable over V . In section 5, we get this result from the hypothesis that hyp-sharps are definably rich.
If somth(V, T )+ is so complicated, how can there ever be a first-order characteriza- tion over V of the T ’s that are satisfiable in some strong outer model? To the point, how can the second sentence in conclusion (1) of Theorem 5.1 be true?
Note that the predicate symbol V occurs in the sentences on the left-hand-side of line (2.7) that witness somth(V, T )+ is too complicated to be definable over V . We shall see in section 3 that if Hyp(V ) satisfies that V -measurable cardinals are definably stationary in V ∩OR, then somth(V, T )+ is a conservative extension of womth(V, T )+. In section 5, we get this result from an analogous hypothesis regarding hyp-sharps. The point is that with this additional non-minimality hypothesis somth(V, T )+ is consistent whenever womth(V, T )+ is consistent. The former is too complicated to be definable over V . But the latter is definable over V , provided that V is a sufficiently non-minimal model of ZFC+. This is how we get the second sentence in conclusion (1) of the main theorem.
A second consequence of line (2.7) is that for a theorem like the main theorem to hold, it is crucial that we do not allow the predicate symbol V to occur in T . Theories T in which V occurs only make sense in the context of strong outer models because weak outer models do not interpret V .
Proposition 2.9. There is no first-order formula good such that for first-order sentences of the language {∈, V } ∪ { a : a ∈ V }
V ² good[] iff is satisfiable in some strong outer model of V
Proof: If is first-order in the language {∈}∪{ a : a ∈ V }, then V is satisfiable in a strong outer model iff V ² .
The final proposition in this section nails down the complexity of the strong outer model theory. Assuming that ∞ = V ∩ OR is definably regular in Hyp(V ), it shows that, unlike womth(V, T )+, there is no simpler definition of somth(V, T )+ than the one we have given.
12
Proposition 2.10. Assume that ∞ is definably regular in Hyp(V ). Then
(1) somth(V, ∅)+ is Σ1 definable over Hyp(V ), consequently
(2) (V, somth(V, ∅)+) ² ZFC.
(3) ΣHyp(V ) m (Hyp(V )) ∩ P(V ) = Σ(V,somth(V,∅)+)
m (V ), for natural numbers m > 1.
This falls just short of being a Σ1 master code in Jensen’s fine-structure terminology. A master code A ⊆ V satisfies
ΣHyp(V ) m+1 (Hyp(V )) ∩ P(V ) = Σ(V,A)
m (V ).
Proof: We already know (1) and (2). Once we have (3) for m = 1, it is rou- tine to get it for all m > 1, using that Hyp(V ) is Σ1 projectible into V to get ΣHyp(V )
m (Hyp(V )) ∩ P(V ) ⊆ Σ(V,somth(V,∅)+) m (V ).
Let us concentrate on the case m = 1. Say that C ⊆ ∞ is a hyp-tower if C is a V -amenable club class of strong limit
cardinals such that (V,C) ² ZFC and, for κ < λ in C, we have that Hyp(Vκ) is the transitive collapse of the Σ1 Skolem hull of Vκ ∪ {Vλ} in Hyp(Vλ). In this case, the inverse of the transitive collapse is a Σ1 elementary map
jκλ: Hyp(Vκ) →Σ1 Hyp(Vλ)
such that jκλ ¹Vκ = id ¹Vκ and jκλ(Vκ) = Vλ. Using these embeddings, let
AC = lim−→ κ∈C Hyp(Vκ)
be the direct limit of this system. Identify wf(AC), the well-founded part of AC , with a transitive set. Now
• AC ² KP; • Hyp(V ) ⊆ wf(AC), since V ∈ wf(AC); and, in fact, • Hyp(V ) = wf(AC), because AC satisfies that there does not exist an admissible
set with V as an element. It is possible to define Σ1 satisfaction for AC over (V ;C) is the following sense: Let
) .
Using that ∞ is definably regular in Hyp(V ), there exists a hyp-tower C such that AC = Hyp(V ). Using this, it follows that if is Σ1 in the language of set theory, then
Hyp(V ) ² [V, a] iff AC ² [V, a], for all hyp-towers C.
Now we come to the main idea for the proof, namely, to reduce quantification over hyp-towers to quantification over strong outer models in the following way: Let C be a hyp-tower. Then (V ;C) ² ZFC. Using Jensen coding, we can force to get a
13
2.4. THE COMPLEXITY OF SOMTH(V, ∅)+
(strong) outer model W = L[x], where x is a real. It is easy to arrange that also Decode(x) = C, where “Decode(x)” is a class definable over L[x].
Let θ(pq, a) be a first-order sentence in {∈, V } ∪ { a : a ∈ V } formalizing this statement:
For every real x, if ∀z (z ∈ L[x]) and C = Decode(x) is a hyp-tower over V , then Hyp(Vκ) ² [Vκ, a], for all sufficiently large κ ∈ C.
Then Hyp(V ) ² [V, a] iff somth(V, ∅)+ ` θ(pq, a).
Finally, note that the right-hand-side of this equivalence is a Σ(V,somth(V,∅)+) 1 (V ) prop-
erty because deductions from somth(V, T )+ lie in V .
3. Global definability of theories satisfiable in outer models
For the purposes of this section, set
ZFC+ = ZFC + “the class of measurable cardinals is definably stationary”.
This just means that every definable closed unbounded class of ordinals contains a measurable cardinal.
The goal of this section is to prove
Theorem 3.1. There exists a parameter-free Π2 formula good(x) in the language of set theory with the following properties.
Let V be a standard transitive model of ZFC+. Assume that κ is a regular un- countable cardinal in V and that T ∈ V is a set of axioms in the language of set theory with parameters in V .
(1) If V ² good[T ], then there exists a weak outer model of V that satisfies T . If also V is sufficiently non-minimal, then there exists a strong outer model of V that satisfies T .
(2) If V ² ¬good[T ] and V is sufficiently non-minimal, then T is not satisfied in any weak (a fortiori, strong) outer model of V .
Sufficient non-minimality. If ∞ is definably regular in Hyp(V ), then V is sufficiently non-minimal for conclusion (2).
More non-minimality is needed for the second sentence of (1). For it, the proof requires that Hyp(V ) satisfies that V -measurable cardinals are definably stationary in ∞. It is open whether extra non-minimality is necessary for (1).
14
3.1. Well-founded direct limits
The hypothesis that ∞ is definably regular in Hyp(V ) secures that the direct limit of a V -definable directed system of well-founded structures is well-founded. Seeing this is a matter of simple admissible set theory, but since this may not be familiar material, let us delay the proof of Theorem 3.1 to prove first a couple of lemmas to this effect.
Lemma 3.2. Assume that ∞ is definably regular in Hyp(V ). (a) Let a ∈ V . If (x, y) is a formula with parameters in Hyp(V ), then
Hyp(V ) ² ∀x∈a ∃y∈V (x, y) → ∃b∈V ∀x∈a ∃y∈b (x, y).
(b) Let a ∈ V . If (x, p) is a formula with parameter p ∈ Hyp(V ), then{ x ∈ a : Hyp(V ) ² (x, p)
} ∈ V.
Proof of (a): Because there exists a function in V from some ordinal α <∞ onto a, we may as well assume that a = α <∞. For γ < α, define in Hyp(V )
f(γ) = the least δ such that ∃y∈Vδ (γ, y).
Because ∞ is definably regular in Hyp(V ), there exists β < ∞ such that Hyp(V ) satisfies ∀x∈α ∃y∈Vβ (x, y).
Proof of (b): Suppose that is Σn and that p ∈ Hyp(V ) is a parameter. Using that ∞ is definably regular in Hyp(V ), there exists κ <∞ and
j: Hyp(Vκ) →Σn Hyp(V )
such that j ¹Vκ = id ¹Vκ; j(Vκ) = V ; a ∈ Vκ; and p ∈ rng(j). Then
{ x ∈ a : Hyp(V ) ² (x, p)
} =
( x, j−1(p)
)} ∈ V.
Lemma 3.3. Assume that ∞ is definably regular in Hyp(V ). Let (Aα, Eα) : α <∞ be a V -definable directed system of well-founded structures with associated embed- dings jαβ : α 6 β <∞. (So Eα is a well-founded binary relation on Aα.) Then the direct limit lim−→ α<∞(Aα, Eα) is well-founded.
Proof: We may as well assume that the embeddings jαβ are all inclusion functions. (Replace a ∈ Aβ with its jαβ-preimage, where α is least such that a ∈ rng(jαβ).) Then lim−→ α<∞(Aα, Eα) ∼= (A,E), where A =
α<∞Aα and E =
α<∞Eα.
Because A and E are V -definable, we know that (A,E) ∈ Hyp(V ). Now the well-founded part of (A,E) is Σ1-definable over Hyp(V ). (For example, see [B], The- orem V.3.1.) Set X = A \ B, where (B,E) is the well-founded part of (A,E). Then X is definable over Hyp(V ).
For a contradiction, suppose that X 6= ∅. Set
α0 = the least α <∞ such that X ∩ Vα 6= ∅ and
αn+1 = the least α <∞ such that Hyp(V ) ² ∀x∈(X ∩ Vαn) ∃y∈(X ∩ Vα) y E x.
15
3.1. WELL-FOUNDED DIRECT LIMITS
To see that this recursion succeeds, note that or any α < ∞, we have X ∩ Vα ∈ V because Vα ∈ V and X is definable over Hyp(V ). Consequently, there does exist an α <∞ such that Hyp(V ) satisfies the sentence ∀x∈(X ∩ Vαn) ∃y∈(X ∩ Vα) y E x.
The sequence αn : n ∈ ω is definable over Hyp(V ), so there exists β < ∞ such that αn < β, for all n, using again that ∞ is definably regular in Hyp(V ). Now X ∩ Vβ ∈ V and ∀x∈(X ∩ Vβ) ∃γ x ∈ Aγ . Hence there exists a δ < ∞ such that X ∩ Vβ ⊆ Aδ. But then (Aδ, Eδ) is ill-founded.
3.2. Proof of the global version of the main theorem With Lemma 3.3 in hand, we turn to the proof of Theorem 3.1. If D is a δ-complete measure on δ, let
Ult(α) ( Hyp(Vδ), D
)
be the αth iterated ultrapower of Hyp(Vδ). Working in V , we can use the associated iterated ultrapower embeddings to define
the direct limit lim−→ α∈V Ult(α)
( Hyp(Vδ), D
) .
Using that ∞ is definably regular in Hyp(V ), this direct limit is well-founded. It follows that
Hyp(M) ∼= lim−→ α∈V Ult(α) ( Hyp(Vδ), D
) ,
where M = lim−→ α∈V Ult(α)(Vδ, D)
is a V -definable inner model of V . This is because both are the smallest admissible set with M as an element.
Lemma 3.4. Assume that V is standard transitive model of ZFC, that ∞ is definably regular in Hyp(V ), and that T ∈ Vδ is a set of axioms in the language of set theory with parameters from Vδ, where δ is a measurable cardinal in V . If womth(Vδ, T )+ is inconsistent, then V does not have a weak outer model satisfying T .
Proof: Assume that womth(Vδ, T )+ is inconsistent. Define in V
M = lim−→ α∈V Ult(α)(Vδ, D),
where D ∈ V is some δ-complete measure on δ. Then ( womth(M,T )+
)lim−→α∈V Ult(α)(Hyp(Vδ),D)
lim−→ α∈V Ult(α)(Hyp(Vδ), D) ∼= Hyp(M),
and so womth(M,T )+ =
( womth(M,T )+
)Hyp(M)
is inconsistent. Now M is a definable inner model of V , so
womth(M,T )+ ⊆ womth(V, T )+
by Lemma 2.6. Consequently womth(V, T )+ is inconsistent. It follows that V does not have a weak outer model satisfying T .
16
3.2. PROOF OF THE GLOBAL VERSION OF THE MAIN THEOREM
Lemma 3.5. Let T ∈ Vκ be a set of sentences in the language of set theory with parameters in Vκ. If κ < δ are measurable, then womth(Vκ, T )+ ⊆ womth(Vδ, T )+.
Proof: Fix a κ-complete measure D on κ. Let
M (α) = Ult(α)(V,D)
be the αth iterated ultrapower of V by D. Let j0α:V → M (α) be the canonical elementary embedding and let κα = j0α(κ). Then κδ = δ. Set M = M (δ) and j = j0δ. Then j(Vκ) = Mδ, where Mδ is the set of elements in M of rank less than δ. Now
j ( Hyp(Vκ)
)
The second equality holds because Hyp(A) = Lα(A), where α is least such that this structure is admissible, henceM correctly calculates Hyp
( j(Vκ)
so womth(Vκ, T )+ ⊆ j
= womth(Mδ, T )+
⊆ womth(Vδ, T )+
The equality above holds on account of the previous displayed sequence of equations, using that womth(N,T )+ is uniformly definable over Hyp(N) from the parameter T . The second containment holds because Mδ is a definable inner model of Vδ.
The V -definable approximation womth(V, T )∗. Define
womth(V, T )∗ =
womth(Vδ, T )+
Lemma 3.6.
(a) If V -measurable cardinals are unbounded in V ∩ OR, then womth(V, T )∗ is deductively closed and womth(V, T ) ⊆ womth(V, T )∗.
(b) If V -measurable cardinals are definably stationary in V ∩OR, then womth(V, T )∗ locally omits ΘV ∩OR. Consequently
womth(V, T )+ ⊆ womth(V, T )∗.
measurable δ<κ with T∈Vδ
womth(Vδ, T )+ `
measurable δ<κ with T∈Vδ
womth(Vδ, T )+ ⊆ womth(Vκ, T )+.
3.2. PROOF OF THE GLOBAL VERSION OF THE MAIN THEOREM
Consequently ∈ womth(Vκ, T )+ ⊆ womth(V, T )∗. To see that womth(V, T ) ⊆ womth(V, T )∗, note by inspection that
womth(V, T ) =
womth(Vδ, T ).
Proof of (b): Work in V . Suppose that womth(V, T )∗ ` ψ(α), for all α ∈ OR. Define f : OR → OR by
f(α) = the least measurable δ such that ψ(α) ∈ womth(Vδ, T )+.
Then f is definable, so there exists a definable club class of γ such that f”γ ⊆ γ. Because V ² ZFC+, there exists a measurable γ in this class. Since womth(Vδ, T )+ ⊆ womth(Vγ , T )+, for measurable δ < γ, it follows that
∀x(x ∈ OR → ψ(x)) ∈ womth(Vγ , T )+.
Hence this formula lies in womth(V, T )∗.
Remark 3.7. If V satisfies ZFC+ and is sufficiently non-minimal, then womth(V, T )∗ = womth(V, T )+.
Proof: We must see that womth(Vδ, T )+ ⊆ womth(V, T )+, for all V -measurable δ. Let D be a δ-complete measure on on δ and let
M = lim−→ α∈V Ult(α)(Vδ, D).
Because V is sufficiently non-minimal, we have that
Hyp(M) ∼= lim−→ α∈V Ult(α) ( Hyp(Vδ), D
) .
womth(M,T )+ ∩ Vδ = womth(Vδ, T )+.
The formula good. Let good(T ) be the formalization of this statement:
T is a set of sentences in the language of set theory with parameters in V and womth(Vδ, T )+ is consistent, for all measurable cardinals δ such that T ∈ Vδ.
Then (1) good(x) is a parameter-free Π2 statement in the language of set theory. (2) Assume that V ² ZFC+.
(a) If V ² good[T ], then womth(V, T )∗ is consistent. By Lemma 3.6 we have womth(V, T )+ is consistent. Hence by Lemma 2.2, T is satisfied in a weak outer model of V .
(b) If V ² ¬good[T ] and V is sufficiently non-minimal, then V does not have a weak outer model that satisfies T by Lemma 3.4.
All that remains to prove in Theorem 3.1 is that if V is a sufficiently non-minimal model of ZFC+ and V ² good[T ], then T is satisfied in some strong outer model of V .
Say that Hyp(V ) satisfies that measurables are definably stationary in ∞ if ∞ is definably regular in Hyp(V ) and, given any Hyp(V )-definable club C ⊆ ∞, there exists a V -measurable κ ∈ C.
18
3.2. PROOF OF THE GLOBAL VERSION OF THE MAIN THEOREM
Lemma 3.8. Assume that Hyp(V ) satisfies that measurables are definably stationary in ∞. For theories T ∈ V , somth(V, T )+ is a conservative extension of womth(V, T )+. Consequently, the former is consistent if and only if the latter is consistent.
Proof: Let be a sentence in the language of set theory with parameters in V and assume that somth(V, T )+ ` . Using our non-minimality hypothesis, let κ be measurable in V with , T ∈ Vκ and such that there exists a Σ1 elementary
j: Hyp(Vκ) → Hyp(V )
with j ¹Vκ = id ¹Vκ and j(Vκ) = V . Then somth(Vκ, T )+ ` . Let D be (in V ) a κ-complete measure on κ and let
MD = lim−→ α∈V Ult(α)(Vκ, D).
Because ∞ is definably regular in Hyp(V ), we have that
Hyp(MD) ∼= lim−→ α∈V Ult(α) ( Hyp(Vκ), D
) .
Hence somth(MD, T )+ ` . Let W ² T be a weak outer model of V . Now D and Vκ are both in V and
V ⊆ W . Hence W is a strong outer model of MD that satisfies T . So W ² , since somth(MD, T )+ ` . Because W was arbitrary, we have that T ²wom . Hence womth(V, T )+ ` .
Question 3.9. Assume just that ∞ is definably regular in Hyp(V ). Is somth(V, T )+ a conservative extension of womth(V, T )+?
4. Hyp-sharps
In this section we develop the theory of indiscernibles for structures of the form (Hyp(Vκ);Vκ, <, a)a∈r, where r ∈ Vκ is transitive and < is a well-ordering of Vκ
(to provide Skolem functions). Following Silver’s development of 0#, we call the blueprints for Ehrenfeucht-Mostowski models that we get “hyp-sharps”. Hyp-sharps, or rather the Ehrenfeucht-Mostowski models they are used to generate, are used in place of measurable cardinals in the proof of Theorem 3.1 to prove the main theorem, Theorem 5.1, in section 5.
Just as in the definition of 0#, in the definition of “hyp-sharp” there are a number of logically simple syntactical requirements, capped by a well-foundedness requirement. We begin with the syntactical requirements.
19
OUTER MODEL SATISFIABILITY
4.1. Remarkable, m-unbounded r-characters Fix a transitive set r. An r-character is a set of first-order formulas in the language Lr that has the following resources:
∈ a binary relation symbol for the membership relation < a binary relation symbol for a well-ordering of Hyp(Vκ) m a constant symbol for Vκ
a, for a ∈ r constant symbols for elements of r
Our convention is that the actual variables of our language are vi, for 1 6 i < ω. We use x, y, and so forth, as variables for these variables.
A remarkable, m-unbounded r-character is a maximal consistent set of for- mulas including
• the diagram of r: a ∈ b, for a ∈ b ∈ r a /∈ b, for a, b ∈ r such that a /∈ b
• a ∈ m, for a ∈ r • “(m,∈) is a transitive model of ZFC” • ∀x∃α x ∈ Lα(m) • KP in the language {∈,m} ∪ { a : a ∈ r } • ∀α Lα(m) 2 KP • “< is a linear order extending the rank preordering,” that is, if rk(x) < rk(y),
then x < y
• “{x : ∀z(z < x↔ z ∈ x) } is closed and unbounded in m under <” • ∀x < m P(x) ∈ m • (Least witnesses)
∀x1 . . . ∀xk
) ,
• (Indiscernibility) (vi1 , . . . , vik
) ↔ (vj1 , . . . , vjk ),
for all increasing sequences of natural numbers i1 < · · · < ik and j1 < · · · < jk and formulas of Lr in which none of the variables vi1 , . . . vik
, vj1 , . . . , vjk are
quantified variables. • (Unboundedness) v1 < m
t(v1, . . . , vk) < m→ t(v1, . . . , vk) < vk+1, for all <-Skolem terms t • (Remarkability)
t(v1, . . . , vk+`) < vk+1 → t(v1, . . . , vk+`) = t(v1, . . . , vk, vk+`+1, . . . , vk+2`),
for all <-Skolem terms t
20
4.1. REMARKABLE, M -UNBOUNDED R-CHARACTERS
<-Skolem terms. In a model of a remarkable, m-unbounded, r-character, there exist definable Skolem functions
f∃x(x,x1,...,xk)(a1, . . . , ak) = {
the <-least b such that [ b, a1, . . . , ak], if such a b exists
∅, otherwise
One approach is to introduce function symbols for these functions into our language and to adjoin axioms defining them. In this conservative extension there are literally Skolem terms t in the language and the last two axiom groups above are literally correct. The usual alternative, which we adopt here, is to treat formulas written with Skolem terms as abbreviations of formulas in the un-Skolemized language, in this case Lr.
4.2. EM(Σ, α) models Suppose that Σ is a remarkable, m-unbounded r-character and that α is a limit ordinal. An EM(Σ, α) model is a structure
A = (A;∈A, <A,mA, aA)a∈r
for Lr, together with a set I ⊆ A satisfying the following: • Finite increasing sequences from I having the same length are indiscernible
in A. • The order type of (I,<A) is α. • (v1, . . . , vk) ∈ Σ iff A ² [i1, . . . , ik], for i1 <A · · · <A ik from I. • A, the universe of A, is the closure of I under the definable Skolem functions
provided by <A. Regarding the last requirement, note that it follows from the previous requirement
that A satisfies the Least Witness axioms, so <A provides definable Skolem functions. Note also that mA and aA, for a ∈ r, are automatically included in the Skolem closure of I because they have names in the language.
A simple application of Compactness is
Lemma 4.1. Let Σ be a remarkable, m-unbounded r-character and let α be a limit ordinal. Then an EM(Σ, α) models exists.
The following is standard model theory:
Lemma 4.2. Let Σ be a remarkable, m-unbounded r-character. Let α 6 β be limit ordinals and let A and B be EM(Σ, α) and EM(Σ, β) models, respectively, as witnessed by sets of indiscernibles I and J , respectively. Let h: I → J be order-preserving. There exists an elementary embedding h:A → B extending h. If h is onto, then h is onto. Consequently, up to isomorphism there exists a unique EM(Σ, α) model, for each limit ordinal α. It follows that if the EM(Σ, α) model is well-founded, then there exists a unique transitive standard EM(Σ, α) model.
Lemma 4.3. Let Σ be a remarkable, m-unbounded r-character, let α be a limit ordi- nal, and let A = (A;∈A, <A,mA, aA)a∈r be the EM(Σ, α)-model, as witnessed by the indiscernibles I.
21
4.2. EM(Σ, α) MODELS
(a) Let β < α be a limit ordinal and let i′ be the βth element of I. Then{ tA(i1, . . . , ik) < mA : i1, . . . , ik ∈ I and i1, . . . , ik <
A i′ }
is exactly the set of <A-predecessors of i′. (b) I is a closed subset of A under <A and is unbounded in mA (but mA /∈ I).
Proof of (a): Suppose first that i1, . . . , ik <A i′ are in I and that tA(i1, . . . , ik) <A
mA. Then by Unboundedness tA(i1, . . . , ik) <A i∗k < A i′, where i∗k be the least element
of I greater than ik. Conversely, suppose that tA(i1, . . . , ik, j1, . . . , j`) <A i′, where i1, . . . , ik <A i′ and
i′ 6A j1, . . . , j`. Because β is a limit ordinal and i′ is the βth element of I, there exist ik+1, . . . , ik+` in I above both ik and tA(i1, . . . , ik, j1, . . . , j`) and below i′. Then
tA(i1, . . . , ik, j1, . . . , j`) = tA(i1, . . . , ik, ik+1, . . . , ik+`) <A i∗k+` < A i′,
using Remarkability. Proof of (b): Closure is a consequence of part (a): If i′ is the βth element of I, where β is a limit ordinal, and if a <A i′, then a = t(i1, . . . , ik), for some i1 <A · · · <A
ik < A i′. And t(i1, . . . , ik) <A i∗k, where, again, i∗k is the least member of I greater
than ik. Unboundedness in mA just uses the Unboundedness axioms: If tA(i1, . . . , ik) <A
mA, then tA(i1, . . . , ik) <A i∗k < A mA.
Of course mA /∈ I because it is defined by the formula v1 = m and elements of I are indiscernible.
Lemma 4.4. Let Σ be a remarkable, m-unbounded, r-character.
(a) The formula ∀x(x < v1 ↔ x ∈ v1) is in Σ.
(b) The formula expressing ∀x < v1 P(x) ∈ v1 is in Σ.
(c) For all parameters a1, . . . , ak ∈ r and formulas (z, x1, . . . , xk) of the language {∈, <}, the formula expressing
∀z ∈ v1 (
(v1,∈, <) ² [z, a1, . . . , ak] ↔ (m,∈, <) ² [z, a1, . . . , ak] )
is in Σ.
Proof: Let A be the EM(Σ, ω1) model, as witnessed by the set of generating indis- cernibles I. It suffices to show that each of these formulas is satisfied by some (hence any) i ∈ I. Part (a). If a ∈A i, then a <A i since <A extends the rank preordering in the sense of A. We must see that there exists i ∈ I such that if a <A i, then a ∈A i, for all a.
It may well be that <A is not a well ordering. Nevertheless, I is a club subset of mA under <A having order type ω1. Also
A ² “{x : ∀z < x z ∈ x } is club in m”.
It follows that there exists some i ∈ I that lies in this set. (Let i0 ∈ I. Let an > A in
be such that an <A mA and has the property that z ∈A an, for all z <A an. Let in+1 >
A an lie in I. If i = supn∈ω in, then i ∈ I, and hence is an element in the universe of A and has the desired property.)
22
4.2. EM(Σ, α) MODELS
Part (b). Let i ∈ I be a limit and let a <A i. Then there exists i1, . . . , ik <A i such that a = tA(i1, . . . , ik). Now a <A i <A mA, so PA(a) ∈A mA. Hence PA(a) <A i′, for some i′ ∈ I. It follows by indiscernibility that PA(a) <A i.
Part (c). Again, assume that i is a limit point of I. Suppose that
A ² “(m,∈, <) ² ∃wψ[b, a1, . . . , ak]”,
where ψ is a formula of the language {∈, <} and a1, . . . , ak ∈ r(Σ) and b ∈A i. Then for some i′ ∈ I,
A ² “(m,∈, <) ² ∃w < i′ ψ[b, a1, . . . , ak]”,
Now b = tA(i1, . . . , in), for some i1, . . . , in <A i, so by indiscernibility,
A ² “(m,∈, <) ² ∃w < i ψ[b, a1, . . . , ak]”.
Part (c) follows, using Tarski’s criterion for elementary substructures.
4.3. Hyp-sharps A hyp-sharp is a remarkable, m-unbounded, r-character Σ for some transitive r, such that the EM(Σ, α) model is well-founded, for all countable α.
Precisely as in Silver’s development of 0#, we have this
Lemma 4.5. Assume that Σ is a remarkable, m-unbounded r-character. The following are equivalent:
• The EM(Σ, α) models is well-founded, for all countable α.
• The EM(Σ, α) model is well-founded for some uncountable α.
• The EM(Σ, α) model is well-founded for all α.
By now the reader’s faith is well tested. It would be good to prove the existence of hyp-sharps.
Lemma 4.6. Assume that κ is Ramsey and that r ∈ Vκ is transitive. There exists a hyp-sharp Σ such that the EM(Σ, κ) model is isomorphic to an elementary substruc- ture of (Hyp(Vκ);∈, Vk, a)a∈r.
Proof: Let < well-order Hyp(Vκ) in such a way that • if rk(a) < rk(b), then a < b and • Vδ is <-least greater than all a ∈ Vδ, for all δ 6 κ.
Note that Vκ is the κth element of Hyp(Vκ) under <, since κ is inaccessible. Note also that {Vδ : δ < κ } is club in Vκ under <. Furthermore, if Vδ 6 z < x < Vδ+1, then rk(z) = rk(x), so z /∈ x. Hence {x : ∀z < x z ∈ x } ∩ Vκ = {Vδ : δ < κ }. Thus {x : ∀z < x z ∈ x } is club in Vκ and includes Vκ itself.
Let I be a set of indiscernibles for A = (Hyp(Vκ);∈, <, Vκ, a)a∈r with order type κ, chosen so that its ωth element is as small as possible.
Let (v1, . . . , vk) ∈ Σ iff A ² [i1, . . . , ik] for i1 < · · · < ik from I. Then Σ is an r-character and the EM(Σ, κ) model is isomorphic to the Skolem hull
of I under the Skolem function for Lr provided by <. This hull is an elementary substructure of (Hyp(Vκ);∈, <, Vκ, a)a∈r.
23
4.3. HYP-SHARPS
We must check that Σ contains the Unboundedness and Remarkability formulas. The Remarkability formulas are handled by Silver’s original argument: Assume that t(i1, . . . , ik+`) < ik+1. Let (i1, . . . , ik) be the first k members of I. Enumerate suc- cessive increasing `-tuples (iδ1, . . . , i
δ `) from I, for δ < κ. Then ik < i01 and iδ` < iγ1 ,
for δ < γ. Let aδ = tA(i1, . . . , ik, iδ1, . . . , i δ `), for δ < κ. By indiscernibility, one of
the following must hold for all δ < γ: aγ < aδ; aγ = aδ; or aδ < aγ . We must rule out the first and last possibilities. The first possibility contradicts that < is a well-ordering. If the last holds, then { aδ : δ < κ } is a set of indiscernibles for A. And aω = tA(i1, . . . , ik, iω1 , . . . , i
ω ` ) < iω1 , which is the ωth element of I. This contradicts
our choice of I. Finally, we turn to Unboundedness. Either I is entirely above Vκ or entirely be-
low Vκ. Now Hyp(Vκ) is projectible into Vκ, so there exists a definable one-to-one function from Hyp(Vκ) into Vκ. If I were entirely above Vκ, then its image under this function would be a set of indiscernibles with a smaller ωth element. So I is entirely below Vκ.
Because I has cardinality κ and Vκ is the κth element of Hyp(Vκ) under <, we have that I is unbounded in Vκ. That the Unboundedness formulas are in Σ follows from this: Suppose that a = tA(i1, . . . , ik) > i∗k, the least element of I greater than ik. By indiscernibility, it follows that a > i, for all i ∈ I. Hence a > Vκ.
Notation r(Σ). Suppose that Σ is a hyp-sharp. Let
r(Σ)
be the transitive set r such that Σ is an r-character, that is, Σ is a complete set of formulas in the language Lr = {∈, <,m} ∪ { a : a ∈ r }.
Notation M α . Suppose that Σ is a hyp-sharp. Let
MΣ α
be the interpretation of m in the transitive standard EM(Σ, α) model. Then • MΣ
α is a transitive standard model of ZFC and r(Σ) ⊆MΣ α .
• Hyp(MΣ α ) is the universe of the EM(Σ, α) model.
• If α < β are limit ordinals, then MΣ α is a rank initial segment of MΣ
β . There exists an elementary embedding
j: Hyp(MΣ α ) → Hyp(MΣ
α and j(MΣ α ) = MΣ
β .
24
OUTER MODEL SATISFIABILITY
4.4. EM(Σ, ∞) models
Assume that Σ is a hyp-sharp. In the same way the EM(Σ, α) model is constructed, we can use Completeness to construct the proper class model EM(Σ,∞), where ∞ is the class of all ordinals. The only divergence from the set case is that, to keep elements of this model—equivalence classes of terms—from themselves being proper classes, we need to use Scott’s trick. Let
AΣ
denote this definable class structure. Let V be a standard transitive set model of ZFC and assume that Σ ∈ V is a hyp-
sharp in the sense of V . In general AΣ, as defined over V , is not actually well-founded, though there is no counterexample to its well-foundedness in V . Nevertheless, working in V ,
MΣ =
is a (genuinely well-founded) inner model of V .
Working outside our standard transitive set model V , let AΣ ∼= AΣ identify the well-found part of AΣ with a transitive set. Since there is no danger of confusion, let us continue to write “AΣ” and “MΣ” for (AΣ)V and (MΣ)V , respectively. Then m bAΣ
= MΣ, and this set is an element of the well-founded part of AΣ. It follows that Hyp(MΣ) is a subset of the well-founded part of AΣ.
Let AΣ α be the transitive standard EM(Σ, α)-model. (So the universe of AΣ
α is Hyp(MΣ
α ).) Then AΣ is (isomorphic to) the direct limit of the AΣ α ’s for α ∈ V ∩OR,
using the embeddings of line (4.1). Consequently, for limit α ∈ V ∩ OR, there exists an elementary embedding
j: AΣ α → AΣ
α and j(MΣ α ) = MΣ.
In passing, we remark that if ∞ is definably regular in Hyp(V ), then by Lemma 3.2, the structure AΣ is well-founded and its universe is equal to Hyp(MΣ). The results proved in this section do not assume that V is sufficiently non-minimal.
In summary, • MΣ is an inner model of V and r(Σ) ⊆MΣ. • Let AΣ be an EM(Σ,∞)-model with its well-founded part identified with a
transitive set. If AΣ is well-founded, then its universe is Hyp(MΣ). In any case Hyp(MΣ) ⊆ wf(AΣ).
• If α < ∞ is a limit ordinal and AΣ α is the EM(Σ, α)-model, then there exists
an elementary embedding j: AΣ
α → AΣ
α and j(MΣ α ) = MΣ.
25
4.4. EM(Σ,∞) MODELS
Lemma 4.7. Suppose that Σ is a hyp-sharp and that r ⊆ r(Σ) is transitive. Let
} .
Then Σ′ is a hyp-sharp.
Proof: Checking the definition, it is easy to see that Σ′ is a remarkable, m-unbounded r-character. Fix any limit α. Let A be the EM(Σ, α) model, as witnessed by the indiscernibles I. Then{
tA(i1, . . . , ik) : i1, . . . , ik ∈ I and only constant symbols a for a ∈ r occur in t }
is the universe of a substructure of A that is isomorphic to the EM(Σ′, α) model. It follows that the EM(Σ′, α) model is well-founded.
5. Local definability of theories satisfiable in outer models In this section, we define what this means and let
ZFC+ = ZFC + “hyp-sharps are definably rich”.
Combining the results on hyp-sharps from section 4 with the results on outer model theories from section 2 we prove the
Main Theorem 5.1. There exists a formula good(x) in the language of set theory with the following properties.
Let V be a standard transitive model of ZFC+. Assume that κ is a regular un- countable cardinal in V and that T ∈ HV
κ is a set of axioms in the language of set theory with parameters in HV
κ .
(1) If HV κ ² good[T ], then there exists a weak outer model of V that satisfies T .
If also V is sufficiently non-minimal, then there exists a strong outer model of V that satisfies T .
(2) If HV κ ² ¬good[T ] and V is sufficiently non-minimal, then T is not satisfied in
any weak (a fortiori, strong) outer model of V .
The formula good(x) can be taken to be parameter-free Π2. If κ > ω1 and r ∈ Hκ
is uncountable, then good(x) can be taken to be Π1 in the parameter r.
Sufficient non-minimality. If ∞ is definably regular in Hyp(V ), then V is sufficiently non-minimal for conclusion (2). In fact, it suffices for (2) just to have that the EM(Σ,∞)-model is well-founded, for all hyp-sharps Σ in the sense of V .
More non-minimality is needed for the second sentence of (1). For it, the proof requires that Hyp(V ) satisfies that hyp-sharps are “definably rich” in V . This holds if Hyp(V ) satisfies that Ramsey cardinals are definably stationary in ∞.
The V -definable approximation womth(V, T )∗. Set
womth(V, T )∗ =
hyp−sharps Σ such that T∈r(Σ); limit ordinals α
womth(MΣ α , T )+.
Note that “womth(V, T )∗ is consistent” can be formalized as a parameter-free first- order formula with a free variable for T .
26
5. LOCAL DEFINABILITY OF THEORIES SATISFIABLE IN OUTER MODELS
Lemma 5.2. Let α be a limit ordinal. Assume that T is a set of first-order axioms in the language of set theory, perhaps with parameters. Assume that Σ and Σ′ are hyp-sharps with T ∈ r(Σ)∩ r(Σ′) and that Σ ∈MΣ′
α . Let δ be a limit ordinal in MΣ′ α .
Then womth(MΣ
Proof: Let β = OR ∩MΣ′ α . Then MΣ
β is a definable inner model of MΣ′ α . Indeed
MΣ β is (MΣ)MΣ
Let j: Hyp(MΣ δ ) → Hyp(MΣ
β ) be elementary and such that j ¹MΣ δ = id ¹MΣ
δ and j(MΣ
womth(MΣ δ , T )+ = j”womth(MΣ
δ , T )+ ⊆ womth(MΣ β , T )+.
An example of a standard transitive model fulfilling the hypotheses of the next lemma is Vκ, where κ is an inaccessible limit of Ramsey cardinals.
Lemma 5.3. Assume that ∞ is definably regular in Hyp(V ) and that hyp-sharps are unbounded in V . That is, assume that for every transitive set r, there exists a hyp- sharp Σ such that r = r(Σ). If womth(V, T )∗ is inconsistent, then womth(V, T )+ is inconsistent.
Proof: Assume that womth(V, T )∗ is inconsistent. Let r be a transitive set large enough that
hyp−sharps Σ∈r such that T∈r(Σ); limit ordinals α∈r
womth(MΣ α , T )+
is inconsistent. Let Σ be a hyp-sharp such that r = r(Σ) and let α /∈ r be a limit ordinal. Then
hyp−sharps Σ∈r
womth(MΣ α , T )+ ⊆ womth(MΣ
α , T )+,
so womth(MΣ α , T )+ is inconsistent. Because there exists an elementary
j: Hyp(MΣ α ) → AΣ
α and j(MΣ α ) = mA,
( womth(mA, T )+
is inconsistent.
Because ∞ is definably regular in Hyp(V ), we have by Lemma 3.2 that AΣ is well- founded and has universe Hyp(MΣ). Hence
( womth(mA, T )+
= womth(MΣ, T )+,
and so womth(MΣ, T )+ is inconsistent. Since MΣ is a definable inner model of V , it follows that womth(V, T )+ is inconsistent.
27
If x is a set, let tc(x)
be the transitive collapse of x. Beware that “TrCl(x)” indicates the transitive closure of x.
Say that a class R ⊆ V is definably rich if, for each n < ω and each a ∈ V , there exists x ∈ V such that
a ⊆ x Σn V
and tc(x) ∈ R. Of course definably rich is not first-order. However, if R is described by a first-
order formula, then “R is definably rich” can be enforced with a scheme of first-order axioms.
A related notion is “definably stationary.” Let X be a definable transitive class. (The only cases of interest here are X = OR and X = V .) Say that S ⊆ X is definably stationary in X, if S∩C 6= ∅, for all closed unbounded (club) C ⊆ X that are definable over V , perhaps with parameters. A class C is unbounded if, given any z ∈ X, there exists x ∈ X such that z ⊆ x; it is closed if, given any z ⊆ C that lies in V , we have z ∈ C. It is equivalent to say that R is definably rich and to say that {x ∈ V : tc(x) ∈ R }
is definably stationary in V .
Before going on, it might be useful to make a few remarks about richness. Definably rich is intermediate in strength between definably stationary and unbounded:
Proposition 5.4.
(a) If R is definably rich in V , then R is unbounded.
(b) U = {x : x is transitive and x ∩OR is not a limit ordinal } is unbounded but not definably rich.
(c) If S is definably stationary in V , then S is definably rich in V .
(d) R = {x : x is transitive and x ∩OR is not a cardinal } is definably rich, but not definably stationary.
Proof: Part (a) is evident from the definition. For (b), observe that if z Σ1 V is non-empty then z ∩OR 6= ∅ and is closed under
ordinal successors, so tc(z) ∩OR is a limit ordinal. For (c), note that
}
is club and definable over V . If x ∈ C ∩ S, then tc(x) = x and a ⊆ x Σn V . For (d), note first that {x : x is transitive and x ∩OR is a cardinal } is club, so R
is definably non-stationary. Now fix n < ω and a ∈ V . We may assume that n > 1 and that a is infinite. Choose z ∈ V such that a∪ |a| ∪ {|a|} ⊆ z Σn V and |z| = |a|. Let x = tc(z). Then |a| < x ∩OR < |a|+, so x ∈ R.
28
5.1. RICHNESS
Let us say that hyp-sharps are definably rich to mean that
{ MΣ
α : Σ is a hyp-sharp and α is a limit ordinal }
is a definably rich class. Before going on, it is worth checking that it is possible for hyp-sharps to be definably
rich.
Lemma 5.5. Suppose that κ is inaccessible and that the set of Ramsey cardinals less than κ is stationary in κ. Then Vκ satisfies ZFC + “hyp-sharps are definably rich”.
Proof: Fix a ∈ Vκ. Let δ < κ be Ramsey and such that a ∈ Vδ Vκ. Let Σ be a hyp-sharp such that a ∈ r(Σ) and the EM(Σ, δ) model is isomorphic to an elementary substructure (
X;∈, Vδ, a ) a∈r
( Hyp(Vδ);∈, Vδ, a
.
Let x = X ∩ Vδ. Then a ⊆ x Vδ Vκ and MΣ δ = tc(x).
5.2. Completing the proof
Lemma 5.6.
(a) If hyp-sharps are unbounded in V , then womth(V, T )∗ is deductively closed and womth(V, T ) ⊆ womth(V, T )∗.
(b) If hyp-sharps are definably rich in V , then womth(V, T )∗ locally omits ΘV . Hence
womth(V, T )+ ⊆ womth(V, T )∗.
Proof of (a): Suppose first that womth(V, T )∗ ` . Let r be a sufficiently large transitive set such that
hyp−sharps Σ∈r
womth(MΣ α , T )+ ` .
Let Σ be a hyp-sharp with r(Σ) = r and let β /∈ r be a limit ordinal. Then
hyp−sharps Σ∈r
womth(MΣ α , T )+ ⊆ womth(MΣ
so womth(MΣ β , T )+ ` . It follows that ∈ womth(MΣ
β , T )+ ⊆ womth(V, T )∗. To see that womth(V, T ) ⊆ womth(V, T )∗, note by inspection that
womth(V, T ) =
hyp−sharps Σ such that T∈r(Σ); limit ordinals α
womth(MΣ α , T ).
5.2. COMPLETING THE PROOF
Proof of (b): Suppose that womth(V, T )∗ ` ψ(β), for every ordinal β. Fix n < ω large enough that
∀β ∃Σ′ ∃δ ( δ is a limit ordinal and T ∈ r(Σ′) and ψ(β) ∈ womth(MΣ′
δ , T )+ )
is Πn in the parameter T . (In fact, n = 2 suffices, but there is no call for parsimony here.) Let Σ be a hyp-sharp and α a limit ordinal such that, for some x, MΣ
α = tc(x) and
TrCl({T}) ⊆ x Σn V.
For each β ∈ MΣ α ∩ OR, there exists a hyp-sharp Σ′ ∈ MΣ
α with T ∈ r(Σ′) and a limit ordinal δ ∈ MΣ
α ∩ OR such that ψ(β) ∈ womth(MΣ′ δ , T )+. It follows that
ψ(β) ∈ womth(MΣ α , T )+, for all β ∈MΣ
α ∩OR. Hence
Lemma 5.7. The following are equivalent:
(a) womth(V, T )∗ is consistent.
(b) For all hyp-sharps Σ with r(Σ) = TrCl({T}), the first-order sentence formal- izing “ womth(m,T )+ is inconsistent” is not in Σ.
Proof: If (b) is false, then there exist Σ and α such that ( Hyp(MΣ
α );MΣ α ,∈, a
) a∈r
satisfies “womth(m,T )+ is inconsistent”. But then womth(MΣ α , T )+ is inconsistent.
Consequently womth(V, T )∗ is inconsistent. Conversely, assume that womth(V, T )∗ is inconsistent. By amalgamating hyp-
sharps as in the proof of the previous lemma, if necessary, there exists a hyp-sharp Σ and a limit ordinal α such that womth(MΣ
α , T )+ is inconsistent. Because this set of formulas is (uniformly) definable over Hyp(MΣ
α ) from the parameters T and MΣ α , a
sentence formalizing “womth(m,T )+ is inconsistent”
} .
And r(Σ′) = TrCl({T}). The formula good(x). Let us say that T is good if
T is a set of axioms in the language of set theory, perhaps with parame- ters, and “womth(m,T )+ is inconsistent” /∈ Σ, for all hyp-sharps Σ such that r(Σ) = TrCl({T}).
Our next task is to calculate the logical complexity of good.
30
5.2. COMPLETING THE PROOF
Lemma 5.8. Assume that hyp-sharps are unbounded in V . Let κ be a regular un- countable cardinal.
(a) If T ∈ Hκ is a set of axioms in the language of set theory, perhaps with parameters, then{
Σ ∈ Hκ : Σ is a hyp-sharp such that T ∈ r(Σ) and MΣ ² T }
(5.1)
is uniformly Π1 definable over Hκ from the parameter T . If r ∈ Hκ is uncount- able (so κ > ω1), then the set in line (5.1) is uniformly Σ1 definable from the parameters r and T .
(b) The set of good T ∈ Hκ is Π2 definable (without parameters) over Hκ. If r ∈ Hκ is uncountable (so κ > ω1), then the set of good T ∈ Hκ is Π1
definable over Hκ from the parameter r.
Proof of (a): If is a linear ordering of ω and Σ is a remarkable, m-unbounded r-character, for the purposes of the current lemma, let
A(Σ,)
denote the Ehrenfeucht-Mostowski model defined using Σ and elements of ω ordered by as indiscernibles.
Working in Hκ, we have that Σ lies in the set of line (5.1) if and only if (i) Σ is a remarkable, m-unbounded, r-character;
(ii) T ∈ r(Σ); (iii) for every linear ordering on ω, if (ω,) is a well-ordering, then A(Σ,) is
well-founded; and (iv) ∀ψ ∈ T “m ² ψ” ∈ Σ. The properties is well-founded and is a well-ordering are 1 over Hκ. The quan-
tification over is unbounded when κ = ω1. If κ > ω1, then the quantification in (iii) can be bounded by a parameter. That
parameter is itself Π1 definable in Hκ, but this gives only a 2 definition. Instead, when κ > ω2, replace (iii) with (iii ′) there exists a standard transitive model M of ZFC such that r,Σ ∈ M and
M ² ∀α(“the EM(Σ, α) model is well-founded”). Because r, hence M , is uncountable, we know that M satisfies this sentence if and
only if the EM(Σ, α) model is actually well-founded, for all α.
Proof of (b): Working in Hκ, T is good if
for every Σ ( if Σ is a hyp-sharp and T ∈ r(Σ) and MΣ ² T ,
then “womth(m,T )+ is inconsistent” /∈ Σ ) .
The quantification over Σ is unbounded.
Say that Hyp(V ) satisfies that hyp-sharps are definably rich in V if ∞ is definably regular in Hyp(V ) and, given any Hyp(V )-definable club C ⊆ V , there exists x ∈ C such that tc(x) = MΣ
α , for some hyp-sharp Σ ∈ V and some limit ordinal α <∞.
Finally, we need
5.2. COMPLETING THE PROOF
Lemma 5.9. Assume that Hyp(V ) satisfies that hyp-sharps are definably rich in V . Then for theories T ∈ V , somth(V, T )+ is a conservative extension of womth(V, T )+. Consequently, the former is consistent if and only if the latter is consistent.
Proof: The proof is exactly like that of Lemma 3.7, using a hyp-sharp in the role played by a measurable cardinal there.
Summarizing, we have that (1) good(x) can be taken to be a parameter-free Π2 formula in the language of set
theory, or, if κ > ω1, to be Π1 in the parameter r, where r is any uncountable set in Hκ.
(2) Assume that V ² ZFC + “hyp-sharps are definably rich”. (a) If V ² good[T ], then womth(V, T )∗ is consistent. By Lemma 5.6 we have
womth(V, T )+ is consistent. Hence by Lemma 2.2, T is satisfied in a weak outer model of V . If, in addition, Hyp(V ) satisfies that hyp-sharps are definably rich in V , then T is also satisfiable in a strong outer model by Lemma 5.9.
(b) If V ² ¬good[T ], then womth(V, T )∗ is inconsistent. So womth(V, T )+ is inconsistent by Lemma 5.3, provided that ∞ is definably regular in Hyp(V ). It follows from Lemma 2.2 that V does not have a weak (or strong) outer model satisfying T .
This completes the proof of Theorem 5.1.
A curio
Forthcoming papers uses the machinery and results of this paper to study the ade- quacy of set forcing and get results concerning definable inner models. As a curiosity from our journey to this point, we end with the following application of the main theorem.
Corollary 5.10. Assume that ∞ is definably regular in Hyp(V ). Let κ be regular and uncountable. Suppose that Pδ : δ < λ is an iteration of κ-cc forcing in which direct limits are taken at δ of cofinality at least κ. Let T ∈ Hκ be a set of axioms in the language of set theory with parameters in Hκ. Let cf(δ) > κ. If T is satisfiable in a weak outer model of V P
γ
, for γ unbounded in δ, then T is satisfiable in a weak outer
model of V P δ
. If Hyp(V ) satisfies that hyp-sharps are definably rich in V , then the same holds for strong outer models.
Proof: The non-minimality hypotheses are preserved by set forcing. Note that HV P
δ
κ =
γ
κ because Pδ is the direct limit of κ-cc forcing. For γ < δ, HP γ
κ is a
transitive substructure, hence Σ0-elementary substructure, of HP δ
κ . Because good(T ) is Π2, it is preserved under unions of chains of Σ0-elementary substructures.
32
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[F2] , Internal consistency and the inner model hypothesis, Bulletin of Symbolic Logic 12, no.4, pp. 591–600.
[FWW] S.D. Friedman, P. Welch, and W.H. Woodin, On the consistency strength of the inner model hypothesis, Journal of Symbolic Logic 73, no.2 (2008), pp. 391–400.
[S1] M.C. Stanley, Forcing closed unbounded subsets of ω2 , Jour. Pure Appl. Log. 110 (2001), pp. 23–87.
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[S3] , Forcing closed unbounded subsets of ℵω1+1 (to appear).
Math Department San Jose State San Jose, CA 95192
e-mail: stanley@math.sjsu.edu