Chang-type models of determinacy

Takehiko Gappo(https://sites.google.com/view/takehikogappo/home)
TU Wien
October 24, 2023
RIMS Set Theory Conference 2023, Kyoto/online
This talk is based on several joint works with
Navin Aksornthong, James Holland, Sandra Müller, and Grigor Sargsyan
as well as my ongoing work.

Research supported by Austrian Science Fund (FWF) international grant number I6087.

1/19

Introduction

Results on CDM

Variants of CDM

Conclusion

Infinite games
For any A ⊆ R := ω ω, the Gale-Stewart game Gω (A) with payoff set A is asfollows: two players take turns choosing natural numbers.
I
II

n0

···

n2
n1

n3

···

I wins the game if ⟨ni | i < ω⟩ ∈ A and otherwise II wins. We say that A is
determined if I or II has a winning strategy in Gω (A).
Definition
The Axiom of Determinacy (AD) is the statement that any A ⊆ R isdetermined.
AD is incompatible with ZFC, but determinacy axioms for definable sets ofreals (such as Borel, analytic, projective sets) are compatible with ZFC.
Assuming ω many Woodin cardinals and a measurable cardinal above
them, L(R) |= AD.
2/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Stronger forms of determinacy

We can strengthen the Axiom of Determinacy in many ways.
1

Change the rule of the game.
Play reals. ⇝ ADR .
Play longer. ⇝ Determinacy for games of countable fixed length, etc.

2

Require additional structural property of determinacy models such as ...
Some largeness of the successor of the continuum. DefineΘ = sup{α ∈ Ord | ∃f (f : R ↠ α)}.
The theory “ADR + Θ is regular” is much stronger than ADR in termsof consistency strength.
Having supercompact/strongly compact measures for ω1 .
Universally Baireness of all sets of reals.

... but what motives us to study such stronger forms of determinacy?
3/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Better lower bounds of consistency strength
To make a long story short, the development of inner model theory in the last
30 years suggests that the hierarchy of determinacy axioms is essential tounderstand the hierarchy of large cardinal axioms.
Question
What is the consistency strength of Proper Forcing Axiom (PFA)/Martin’s
Maximum (MM)?
It is known that PFA and MM (and its strengthening MM++ ) are consistent
relative to a supercompact cardinal.
The best known lower bound of consistency strength of PFA is given by Core
Model Induction (CMI), which produces models of strong determinacy axiomsof the form V = L(℘(R)), making heavy use of inner model theory.
The result by Sargsyan–Trang on sealing, however, shows the current
framework of CMI cannot reach the level of Woodin limit of Woodin cardinals.
4/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Better upper bounds of consistency strength
Some fragements of MM++ can be forced over determinacy models, startingfrom weaker assumption than a supercompact cardinal.
Theorem (Woodin)
Assume V = L(℘(R)) + ADR + Θ is regular. If G ⊆ Pmax ∗ Add(ω3 , 1) isV -generic, thenV [G ] |= ZFC + MM++ (c).
Theorem (Larson–Sargsyan)
It is consistent relative to a Woodin limit of Woodin cardinals that there is atransitive model M of ADR + Θ is regular such that if
G ⊆ (Pmax ∗ Add(ω3 , 1) ∗ Add(ω4 , 1))M is M-generic, thenM[G ] |= ZFC + MM++ (c) + ¬□(ω3 ) + ¬□(ω4 ).
It follows that the K c construction breaks down in M[G ].
The above M is a ”Chang-type” model of the form V ̸= L(℘(R)).
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

5/19
TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Outline of this talkWe look for interesting models of determinacy of the form V ̸= L(℘(R)).
The Chang model over the derived model (CDM) is such an example.
We will show two applications of CDM.
We will also talk about variants of CDM highlighted in yellow in the tablebelow:
Solovay
Chang Chang over DM
L(R)
L(R, µ)L(R, µα )L(R, µ∞ )

CM
CM+
CM+
α
CM+
∞

CDM
CDM+
CDM+
α
CDM+
∞

6/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Standard determinacy models: Derived models DM
There is a canonical way to get determinacy model of the form V = L(℘(R))
from large cardinals due to Woodin.
Suppose that λ is a limit of Woodin cardinals and let g ⊆ Col(ω, <λ) be
V -generic. Let[ V [g ∩Col(ω,<α)]
R∗ =
R
α<λ

Γ = {A ⊆ R∗ | L(A, R∗ ) |= AD+ }In V (R∗ ), the derived model of V at λ is defined as DM = L(Γ, R∗ ).
DM |= AD+ .
If λ is also a limit of < λ-strong cardinals, then DM |= ADR .
If ∃κ < λ (κ is λ-supercompact), then DM |= ADR + Θ is regular.
Any model of AD+ + V = L(℘(R)) can be represented as a derived model.
7/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Chang model CM

The Chang model is the minimal inner model of ZF closed under countablesequences. More precisely,[CM =L(ω α).
α∈Ord

Theorem (Woodin)
Suppose that there is a proper class of Woodin limits of Woodin cardinals.
1

The theory of CM cannot be changed by set-sized forcings.

2

CM |= AD+ in any generic extensions.

By Mitchell’s work on the sharp of the Chang model, CM ̸|= ADR .

8/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Chang models over derived models CDM
Assume the following:
Let V be a countable model of ZFC that is a hod mouse.
δ is a regular limit of Woodin cardinals in V.
Let g ⊆ Col(ω, <δ) is V-generic.
Let L(Γ, R∗ ) be the derived model of V at δ.
Theorem (Sargsyan, [Sar21])
In V[g ], there is a transitive model M∞ of ZFC − Power such that M∞ has alargest cardinal δ∞ andL(M∞ , ∪α<δ∞ ω α, Γ, R∗ ) |= AD+ .
This model is called the Chang model over the derived model (CDM).
Actually, by G.–Sargsyan’s result [GS22],
CDM |= ADR + Θ is regular.
9/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Two applications of CDM
Combining CDM with standard genericity iteration argument, we got:
Theorem (G.–Sargsyan, [GS23])
Let M be a countable hod mouse with universally Baire iteration strategy suchthat M |= ZFC+there is a Woodin limit of Woodin cardinals. Then
CM |= AD+in any generic extensions.
Also, the study of the Pmax extension of CDM led to:
Theorem (Aksornthong, G., Holland & Sargsyan, [AGHS23])
Let (⋆)κ be the following statement:
the restriction of the club filter on κ ∩ Cof(ω) to HOD is an ultrafilter in HOD.
Then it is consistent relative to a Woodin limit of Woodin cardinals that
ZFC + (⋆)κ holds for κ ∈ {ω1 , ω2 , ω3 }.
10/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Solovay model L(R, µ)
For A ⊆ ℘ω1 (R), we consider the Gale-Steward game on reals GR (A) withpayoff set A:
I
II

x0

···

x2
x1

x3

···

I wins the game if {xi | i < ω} ∈ A. Then a club filter µ on ℘ω1 (R) can becharacterized byA ∈ µ ⇐⇒ I has a winning strategy for the game GR (A).
The Solovay model is defined as L(R, µ) := L(R)[µ].
Theorem (Woodin)
If there are ω 2 Woodin cardinals and a measurable cardinal above them, then
L(R, µ) |= AD+ + µ is a supercompact measure on ℘ω1 (R).
11/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

First example of generalized Chang models: CM+

Woodin introduced a variant of CM corresponding to the Solovay model:[
CM+ =L(ω α)[µα ],α∈Ordωwhere µα is the club filter on ℘ω1 ( α).
Theorem (Woodin)
If there is a proper class of Woodin limits of Woodin cardinals, then
CM+ |= AD+ + ω1 is supercompact.
(Ikegami & Trang) If ω1 is supercompact, then AD+ is equivalent to ADR .
(Woodin) Assuming the determinacy of some definable game of length ω1 ,which is still unknown to be consistent, CM+ |= Θ is regular.

12/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Chang models over derived models with supercompact measures
Assume the following:
Let V be a countable model of ZFC that is a hod mouse.
δ is a regular limit of Woodin cardinals in V.
Let g ⊆ Col(ω, <δ) is V-generic.
Let L(Γ, R∗ ) be the derived model of V at δ.
Theorem (G.–Müller–Sargsyan, [GMS23])
In V[g ], there is a transitive model M∞ of ZFC − Power such that M∞ has alargest cardinal δ∞ andL(M∞ , ∪α<δ∞ ω α, Γ, R∗ )[⟨µ′α | α < δ∞ ⟩] |= AD+ + ADR + Θ is regular+ ω1 is <δ∞ -supercompact,
where µ′α is the club filter on ℘ω1 (α).
13/19Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

CM+ and CDM+
Independently, Steel found the construction of CDM with club measures on℘ω1 (ω α), assuming that δ is measurable in addition.
By modifying G.–Sargsyan’s proof in [GS23], he showed:
Theorem (Steel, [St23])
Let M be a countable hod mouse with universally Baire iteration strategy suchthat M |= ZFC+ there is a measurable Woodin limit of Woodin cardinals.
Then
CM+ |= AD+ + “ω1 is supercompact”
in any generic extensions.
Very recently, based on [GMS23], I found another construction of CDM withclub measures on ℘ω1 (ω α) without assuming the measurability of δ. We callthis model CDM+ .
The advantage of my construction is that it can be adopted to construct manyother variants of CDM.
14/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Generalized Solovay/Chang models
Fix α < ω1 . Let X α ⊆ α (℘ω1 (R)) be the set of continuous increasing sequenceof countable sets of reals of length α. For A ⊆ X α , let GRα (A) be the game on
reals of length ω α with payoff set A:
I
II

x00

···

x0β

···
x1β

x10

I wins the game if ⟨σ β | β < α⟩ ∈ A, where σ β = {xiγ | γ < β, i < ω}. Define a
filter µα on Xα by
A ∈ µα ⇐⇒ I has a winning strategy for the game GRα (A).
The α-Solovay model is defined as L(R, µα ) = L(R)[µα ].
The corresponding generalized Chang model is defined as[
CM+L(ω β)[µα
α =
β ],
β∈Ord
α
α
ω
where µαβ is a filter on Xβ ⊆ (℘ω1 ( β)) defined in a similar fashion.
15/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Generalized Solovay/Chang models, continued
Let X ∞ ⊆ <ω1 (℘ω1 (R)) be the set of continuous increasing sequence ofcountable sets of reals of countable length. For A ⊆ X ∞ , let GR∞ (A) be thegame on reals of length ω1 :
I
II

x00

···

x0β

···
x1β

x10

I wins the game if ⟨σ β | β < γ⟩ ∈ A for club many γ < ω1 , whereσ β = {xiγ | γ < β, i < ω}. Define a filter µ∞ on Xα by
A ∈ µ∞ ⇐⇒ I has a winning strategy for the game GR∞ (A).
The ∞-Solovay model is defined as L(R, µ∞ ) = L(R)[µ∞ ].
The corresponding generalized Chang model is defined as[
CM+L(ω β)[µ∞
∞ =
β ],
β∈Ord
∞
<ω1
(℘ω1 (ω β)) defined in a similar fashion.
where µ∞β is a filter on Xβ ⊆Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

16/19
TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Generalized Chang models over derived models (ongoing work)
Assume the following:
Let V be a countable model of ZFC that is a hod mouse.
δ is a regular limit of Woodin cardinals in V.
Let g ⊆ Col(ω, <δ) is V-generic.
In V[g ], we can construct a model CDM+
∞ such thatααCDM+α |= ADR + Θ is regular + µβ is an ultrafilter on Xβ for any β < δ∞ .

Moreover, if δ is measurable, we can construct a model CDM+
∞ such that∞∞
CDM+
∞ |= ADR + Θ is regular + µβ is an ultrafilter on Xβ for any β < δ∞ .

These results are proved by combining the technique of [GMS23] with theargument in [Tr13], where Trang introduced generalized Solovay models.
(Generalized Chang models are introduced in [Woo].)
17/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Summary
CDM is the “Chang version” of the derived model, which can satisfy
stronger determinacy axioms than the Chang model does.
CDM can be used to show CM |= AD+ and used as a ground model for
the Pmax forcing to get new consistency results.
Moreover, we can construct variants of CDM corresponding to generalized
Solovay/Chang models.
Solovay

Chang

Chang over DM

L(R)
L(R, µ)L(R, µα )L(R, µ∞ )

CM
CM+
CM+
α
CM+
∞

CDM
CDM+
CDM+
α
CDM+
∞

Thank you!
18/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU WienIntroduction

Results on CDM

Variants of CDM

Conclusion

Main references
Grigor Sargsyan, Covering with Chang models over derived models, Adv. Math. 384 (2021),
Paper No. 107717, 21.
Takehiko Gappo & Grigor Sargsyan, On derived models of self-iterable universes, Proc.
Amer. Math. Soc. 150 (2022), 1321-1329.
Takehiko Gappo & Grigor Sargsyan, Determinacy in the Chang model, available athttps://arxiv.org/abs/2302.06487.
Navin Aksornthong, James Holland, Takehiko Gappo & Grigor Sargsyan, On ω-strongly
measurable cardinals in Pmax extensions, available at https://arxiv.org/abs/2307.13682.
Takehiko Gappo, Sandra Müller & Grigor Sargsyan, Chang models over derived models withsupercompact measures, available at https://arxiv.org/abs/2307.08607.
John R. Steel, Notes on work of Gappo and Sargsyan, available athttps://math.berkeley.edu/~steel/papers/Publications.html.
Nam Trang, Generalized Solovay measures, the HOD analysis, and the core model induction,
PhD thesis, 2013.
W. Hugh Woodin, Determinacy and generic absoluteness, to appear.
19/19
Takehiko Gappo (https://sites.google.com/view/takehikogappo/home)
Chang-type models of determinacy

TU Wien