Inner models from logics and the generic multiverse - Set Theory...

Inner models from logi s and the generi multiverse

Daisuke Ikegami

University of California, Berkeley

Universit�a di Torino

27. September 2013

Beginning: Warning

We work in ZFC + \There are a proper lass of Woodin ardinals".

Goal & Result

Goal

Constru t a model of set theory whi h is \ lose to" HOD but easier to

analyze.

Goal & Result

Goal


analyze.

Theorem

Suppose there is a super ompa t ardinal and assume that the

Conje ture with real parameters holds in any set generi extension.

Goal & Result

Goal


analyze.

Theorem



Then there is a de�nable lass M su h that

1

M is an inner model of ZFC and M � HOD,

Goal & Result

Goal


analyze.

Theorem




1


2

M is invariant under the generi multiverse,

Goal & Result

Goal


analyze.

Theorem




1


2


3

M ontains all the reals in the mi e known to exist so far,

M is losed under all the mouse operators known to exist so far, and

Goal & Result

Goal


analyze.

Theorem




1


2


3

M ontains all the reals in the mi e known to exist so far,

M is losed under all the mouse operators known to exist so far, and

4

M satis�es GCH.

Motivation 1: HOD Conje ture

Theorem (Woodin)

Let � be extendible. Then one of the following holds:

1

for every regular > �, is measurable in HOD, or

2

for every singular ardinal > �, is singular in HOD and

(

+

)

HOD

=

+

.

Motivation 1: HOD Conje ture

Theorem (Woodin)

Let � be extendible. Then one of the following holds:

1

for every regular > �, is measurable in HOD, or

2

for every singular ardinal > �, is singular in HOD and

(

+

)

HOD

=

+

.

De�nition (Woodin)

HOD Conje ture states that the latter ase in the above theorem holds.

Motivation 1: HOD Conje ture td.

1

HOD Conje ture is onne ted to the Inner Model Program for a

super ompa t ardinal.

2

HOD Conje ture has an appli ation to the problem on the existen e

of Reinhardt ardinals in ZF.


1



2



To solve HOD Conje ture, one would expe t a �ne analysis of HOD. But

HOD is very \non-absolute", e.g.,

Proposition (Folklore)

There is a lass partial order whi h for es that V = HOD.


1



2



To solve HOD Conje ture, one would expe t a �ne analysis of HOD. But

HOD is very \non-absolute", e.g.,

Proposition (Folklore)

There is a lass partial order whi h for es that V = HOD.

Goal


analyze.

Motivation 2: Inner Model Theory

Question

What is the limitation of the urrent method of Inner Model Theory?

Motivation 2: Inner Model Theory

Question

What is the limitation of the urrent method of Inner Model Theory?

The keyword is �

2

1

(uB).

Motivation 2: Inner Model Theory; �

2

1

(uB)

De�nition

1

A formula � is �

2

1

(uB) if it is of the form

(9A : universally Baire) (H

!

1

;2;A) � ;

where is a �rst order formula.

2

A set A � H

!

1

is �

2

1

(uB) if it is de�ned by a �

2

1

(uB) formula.

3

A set A � H

!

1

is �

2

1

(uB) in a ountable ordinal if there is a ountable

ordinal � su h that both A and H

!

1

nA are �

2

1

(uB) with parameter �.


2

1

(uB)

De�nition

1


2

1



!

1

;2;A) � ;


2

A set A � H

!

1

is �

2

1


2

1

(uB) formula.

3

A set A � H

!

1

is �

2

1



!

1

nA are �

2

1


Remark

1

All the reals in the mi e known to exist so far are �

2

1

(uB) in a

ountable ordinal.


2

1

(uB)

De�nition

1


2

1



!

1

;2;A) � ;


2

A set A � H

!

1

is �

2

1


2

1

(uB) formula.

3

A set A � H

!

1

is �

2

1



!

1

nA are �

2

1


Remark

1

All the reals in the mi e known to exist so far are �

2

1

(uB) in a

ountable ordinal.

2

If M is A- losed for every A whi h is universally Baire and �

2

1

(uB),

then M is losed under all the mouse operators known to exist so far.

Motivation 2: Inner Model Theory; A- losure

De�nition (A- losure)

Let A be universally Baire. An !-model M of ZFC is A- losed if for any

V -generi �lter G on a partial order in M,

M[G ℄ \ A

V [G ℄

2 M[G ℄:





M[G ℄ \ A

V [G ℄

2 M[G ℄:

Example

1

For an !-model M of ZFC, the following are equivalent:

M is A- losed for any �

1

1

-set A, and

M is well-founded.





M[G ℄ \ A

V [G ℄

2 M[G ℄:

Example

1



1

1

-set A, and

M is well-founded.

2


1

M is A- losed for every �

1

2

-set A, and

2

M � \ every set has a sharp".





M[G ℄ \ A

V [G ℄

2 M[G ℄:

Example

1



1

1

-set A, and

M is well-founded.

2


1

M is A- losed for every �

1

2

-set A, and

2

M � \ every set has a sharp".

3

The more ompli ated A is, the more mi e M has if M is A- losed.

Result; restated

Theorem




Result; restated

Theorem




1

M is a transitive proper lass model of ZFC and M � HOD,

Result; restated

Theorem




1


2


Result; restated

Theorem




1


2


3

M ontains all the reals whi h are �

2

1

(uB) in a ountable ordinal,

M is A- losed for every A whi h is �

2

1

(uB) and universally Baire, and

Result; restated

Theorem




1


2


3

M ontains all the reals whi h are �

2

1

(uB) in a ountable ordinal,

M is A- losed for every A whi h is �

2

1

(uB) and universally Baire, and

4

M satis�es GCH.

Inner models from logi s

De�nition

Given a logi L,

L

0

(L) = ;;

L

�+1

(L) = Def

L

�

(L

�

(L);2)

�

;

L

(L) =

[

�<

L

�

(L) ( is limit);

L(L) =

[

�2On

L

�

(L):


De�nition

Given a logi L,

L

0

(L) = ;;

L

�+1

(L) = Def

L

�

(L

�

(L);2)

�

;

L

(L) =

[

�<

L

�

(L) ( is limit);

L(L) =

[

�2On

L

�

(L):

Example

When L is �rst order logi , L(L) is L.

When L is full se ond (or higher) order logi , L(L) is HOD.

Kennedy, Magidor, and V�a�an�anen explored on inner models from �rst

order logi with generalized quanti�ers.


De�nition

Given a logi L,

L

0

(L) = ;;

L

�+1

(L) = Def

L

�

(L

�

(L);2)

�

;

L

(L) =

[

�<

L

�

(L) ( is limit);

L(L) =

[

�2On

L

�

(L):

Example

When L is �rst order logi , L(L) is L.

When L is full se ond (or higher) order logi , L(L) is HOD.

Kennedy, Magidor, and V�a�an�anen explored on inner models from �rst

order logi with generalized quanti�ers.

In this we talk, we will use Woodin's -logi for a desired inner model M.

Inner models from logi s: the generi multiverse

De�nition

1

Let M;N be transitive models of ZFC. Then M is a ground model of

N if N is a set generi extension of M.


De�nition

1



2

The generi multiverse of V (denoted by M (V )) is the smallest

olle tion of models of ZFC ontaining V and losed under taking set

generi extensions and grounds.


De�nition

1



2

The generi multiverse of V (denoted by M (V )) is the smallest

olle tion of models of ZFC ontaining V and losed under taking set

generi extensions and grounds.

Theorem (Woodin)

There is a re ursive translation � 7! �

�

of �rst order formulas su h that

the following are equivalent:

1

�

�

is true in V , and

2

M � � for every M 2 M (V ).

Inner models for logi s: the model L

De�nition

Let � be a �

2

formula and be a �

2

formula in the language of set theory.

We say (�; ) is a �

ZFC

2

-pair if

ZFC ` \(8~x) �(

~x)$ (

~x):

Inner models for logi s: the model L

De�nition

Let � be a �

2

formula and be a �

2

formula in the language of set theory.

We say (�; ) is a �

ZFC

2

-pair if

ZFC ` \(8~x) �(

~x)$ (

~x):

De�nition

Let A be a �rst-order stru ture in

T

M (V ),~a 2 A

<!

, and (�; ) be a

�

ZFC

2

-pair.

Then the triple (�; ;~a) is suitable to A if for any element x of A, either

(8M 2 M (V )) M � �[x ;~a;A℄

or

(8M 2 M (V )) M � :�[x ;~a;A℄.

Inner models from logi sw: the model L

td.

De�nition

1

Let (�; ;~a) be suitable to A. Then a set X � A is -de�nable via

(�; ;~a) if X = fx 2 A j (8M 2 M (V )) M � �[x ;

~a;A℄g.


td.

De�nition

1


(�; ;~a) if X = fx 2 A j (8M 2 M (V )) M � �[x ;

~a;A℄g.

2

Def

(A) is the olle tion of -de�nable subset of A via some (�; ;~a)

suitable to A.


td.

De�nition

1


(�; ;~a) if X = fx 2 A j (8M 2 M (V )) M � �[x ;

~a;A℄g.

2

Def

(A) is the olle tion of -de�nable subset of A via some (�; ;~a)

suitable to A.

De�nition

L

0

= ;;

L

�+1

= Def

�

(L

�

;2)

�

;

L

=

[

�<

L

�

( is limit);

L

=

[

�2On

L

�

:

The model L

; Basi properties

Observation

L

= (L

)

M

for all M 2 M (V ).

The model L

; Basi properties

Observation

L

= (L

)

M

for all M 2 M (V ).

Observation

L

� HOD.

The model L

; Basi properties

Observation

L

= (L

)

M

for all M 2 M (V ).

Observation

L

� HOD.

Proposition

L

is an inner model of ZF.

Point: Def

(A) � Def

FOL

(A).

The model L

; Useful lemma and AC

Let G (V ) be the olle tion of all set generi extensions of V .

One an de�ne L

using G (V ) instead of M (V ).

The model L



One an de�ne L


Lemma

\L

in G (V )" = \L

in M (V )".

The model L



One an de�ne L


Lemma

\L

in G (V )" = \L

in M (V )".

Proof.

In bla kboard?

The model L



One an de�ne L


Lemma

\L

in G (V )" = \L

in M (V )".

Proof.

In bla kboard?

Theorem


-Conje ture with real parameters holds in any set generi extension.

Then AC holds in L

.

Ba kground 2; -logi

De�nition

Let � be a �

2

senten e with real parameters in the language of set theory.

1

We say � is -valid if � is true in any set generi extension.

Ba kground 2; -logi

De�nition

Let � be a �

2


1


2

We say � is -provable if there is a universally Baire set A su h that if

M is a ountable transitive model of ZFC and A- losed, then M � �.

Ba kground 2; -logi

De�nition

Let � be a �

2


1


2



De�nition

-Conje ture with real parameters states that � is -valid i� � is

-provable for all �.

Ba kground 2; -logi

De�nition

Let � be a �

2


1


2



De�nition

-Conje ture with real parameters states that � is -valid i� � is

-provable for all �.

Remark

-Conje ture with real parameters holds in any set generi extension of a

anoni al inner model with a proper lass of Woodin ardinals.

The model L

; Coming ba k to AC

Theorem



Then AC holds in L

.

The model L

; Coming ba k to AC

Theorem



Then AC holds in L

.

Points:

1

Using -Conje ture, one an redu e the omplexity of Def

to

�

2

1

(uB) in suÆ iently large generi extensions.

2

Using a super ompa t ardinal, one an prove that uB = Æ

+

0

-uB (Æ

0

is

the �rst Woodin ardinal) in suÆ iently large generi extensions.

Using these points, L

an ompute its anoni al well-order on L

�

for all

�.

The model L

; Maximality

Theorem



Then

1

L

ontains all the reals whi h are �

2

1

(uB) in a ountable ordinal, and

2

L

is A- losed for every A whi h is �

2

1

(uB) and universally Baire.

The model L

; Maximality

Theorem



Then

1

L

ontains all the reals whi h are �

2

1

(uB) in a ountable ordinal, and

2

L

is A- losed for every A whi h is �

2

1

(uB) and universally Baire.

Points:

1

Using a super ompa t ardinal, one an prove that uB = Æ

+

0

-uB (Æ

0

is

the �rst Woodin ardinal) in suÆ iently large generi extensions.

2

Then one an redu e �

2

1

(uB) to a simpler statement that Def

an

ompute.

The model L

; GCH

Theorem



Then L

satis�es GCH.

The model L

; GCH

Theorem



Then L

satis�es GCH.

For CH in L

:

Lemma

The reals in L

are exa tly those whi h are �

2

1

(uB) in a ountable ordinal.

Moreover, in V , there is a good �

2

1

(uB) well-order of the reals in L

.

The model L

; GCH

Theorem



Then L

satis�es GCH.

For CH in L

:

Lemma

The reals in L


2

1



2

1


.

Then using the s ale property of �

2

1

(uB), for a given x in L

, one an pi k

a real in L

whi h odes the initial segment of x w.r.t. this well-order.

The model L

; GCH

Theorem



Then L

satis�es GCH.

For CH in L

:

Lemma

The reals in L


2

1



2

1


.

Then using the s ale property of �

2

1

(uB), for a given x in L

, one an pi k

a real in L

whi h odes the initial segment of x w.r.t. this well-order.

For 2

�

= �

+

in L

:

Prove CH in (L

)

Coll(!;�)

in the same way.

Questions

Is L

lose to HOD, i.e., an L

ompute the su essors of suÆ iently

large singular ardinals orre tly as in HOD?

Questions

Is L




The answer is \No" if Hamkins' Ne essity Maximality Prin iple holds.

Questions

Is L





Theorem (Woodin)

If ZF + AD

R

+ \� is regular" is onsistent, then so is ZFC + Ne essity

Maximality Prin iple + \There are a proper lass of Woodin ardinals".

Questions

Is L





Theorem (Woodin)

If ZF + AD

R



Is Ne essity Maximality Prin iple onsistent with a super ompa t ardinal?

Questions

Is L





Theorem (Woodin)

If ZF + AD

R



Is Ne essity Maximality Prin iple onsistent with a super ompa t ardinal?

What kind of large ardinals an L

an a ommodate? How about

measurable ardinals, super ompa t ardinals et .?

Inner models from logics and the generic multiverse - Set Theory...

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