Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Ideals and determinacy

Nam Trang
University of North Texas
Fall 2023 RIMS Set Theory workshop
October 25, 2023

Research supported by NSF CAREER grant DMS-1945592

Nam Trang

Ideals and determinacy

1 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Motivating problems

We are motivated by the following long standing open conjecture concerning the following
principles.

Nam Trang

Ideals and determinacy

2 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Motivating problems

We are motivated by the following long standing open conjecture concerning the following
principles.
Conjecture (H.W. Woodin, [Woo10])
The following theories are equiconsistent.
1

ADR +Θ is regular.

2MM(c)+“JNS is weakly presaturated. "3

CH+NS ω1 S is ω1 dense for densely many stationary S.

Nam Trang

Ideals and determinacy

2 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Motivating problems

We are motivated by the following long standing open conjecture concerning the following
principles.
Conjecture (H.W. Woodin, [Woo10])
The following theories are equiconsistent.
1

ADR +Θ is regular.

2MM(c)+“JNS is weakly presaturated. "3

CH+NS ω1 S is ω1 dense for densely many stationary S.

This is Problem 12 of [Woo10].

Nam Trang

Ideals and determinacy

2 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Motivating problems

We are motivated by the following long standing open conjecture concerning the following
principles.
Conjecture (H.W. Woodin, [Woo10])
The following theories are equiconsistent.
1

ADR +Θ is regular.

2MM(c)+“JNS is weakly presaturated. "3

CH+NS ω1 S is ω1 dense for densely many stationary S.

This is Problem 12 of [Woo10].
In this talk, we examine variations of hypothesis (3) and show that they are indeed equiconsistent
with (1).

Nam Trang

Ideals and determinacy

2 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Motivating problems

We are motivated by the following long standing open conjecture concerning the following
principles.
Conjecture (H.W. Woodin, [Woo10])
The following theories are equiconsistent.
1

ADR +Θ is regular.

2MM(c)+“JNS is weakly presaturated. "3

CH+NS ω1 S is ω1 dense for densely many stationary S.

This is Problem 12 of [Woo10].
In this talk, we examine variations of hypothesis (3) and show that they are indeed equiconsistent
with (1).

Nam Trang

Ideals and determinacy

2 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.
• We assume I is a proper, countably complete, normal.

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.
• We assume I is a proper, countably complete, normal.
• I is c-dense if PI has a dense set of size c.
• Let G ⊂ PI be generic and let jG : V → Ult(V , G ) be the associated generic ultrapower.

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.
• We assume I is a proper, countably complete, normal.
• I is c-dense if PI has a dense set of size c.
• Let G ⊂ PI be generic and let jG : V → Ult(V , G ) be the associated generic ultrapower.
• We say that I is precipitous if whenever G , jG etc. are as above, Ult(V , G ) is well-founded.

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.
• We assume I is a proper, countably complete, normal.
• I is c-dense if PI has a dense set of size c.
• Let G ⊂ PI be generic and let jG : V → Ult(V , G ) be the associated generic ultrapower.
• We say that I is precipitous if whenever G , jG etc. are as above, Ult(V , G ) is well-founded.
• We say that I on Pω1 (c) is strong if it is precipitous and whenever G , jG etc. are as above,jG (ω1 ) = c+ .

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.
• We assume I is a proper, countably complete, normal.
• I is c-dense if PI has a dense set of size c.
• Let G ⊂ PI be generic and let jG : V → Ult(V , G ) be the associated generic ultrapower.
• We say that I is precipitous if whenever G , jG etc. are as above, Ult(V , G ) is well-founded.
• We say that I on Pω1 (c) is strong if it is precipitous and whenever G , jG etc. are as above,jG (ω1 ) = c+ .
• We say that I on Pω1 (c) is pseudo-homogeneous if whenever G , jG etc are as above, for allα ∈ ON, s ∈ ON ω , for all λ < c+ , θ is a formula in the language of set theory, the statement
Ult(V , G )  θ[α, jG (s), jG [λω ]]is independent of the choice of G .

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorem• For any set X , let Pω1 (X ) be the set of countable subsets of X . Let I be an ideal on Pω1 (X )or on ω1 . We write I + for the collection of I-positive sets and FI for the dual filter of I.
Let PI = P(Pω1 (X ))/I (P(ω1 )/I) be the corresponding boolean algebra induced by I.
• We assume I is a proper, countably complete, normal.
• I is c-dense if PI has a dense set of size c.
• Let G ⊂ PI be generic and let jG : V → Ult(V , G ) be the associated generic ultrapower.
• We say that I is precipitous if whenever G , jG etc. are as above, Ult(V , G ) is well-founded.
• We say that I on Pω1 (c) is strong if it is precipitous and whenever G , jG etc. are as above,jG (ω1 ) = c+ .
• We say that I on Pω1 (c) is pseudo-homogeneous if whenever G , jG etc are as above, for allα ∈ ON, s ∈ ON ω , for all λ < c+ , θ is a formula in the language of set theory, the statement
Ult(V , G )  θ[α, jG (s), jG [λω ]]is independent of the choice of G .

Nam Trang

Ideals and determinacy

3 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Definition (HI and DI)• DI is the statement: CH holds and there is an ω1 -dense ideal on ω1 .

Nam Trang

Ideals and determinacy

4 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Definition (HI and DI)• DI is the statement: CH holds and there is an ω1 -dense ideal on ω1 .
• HI is the statement: CH holds and there is an ω1 -dense ideal I on ω1 such that lettingg ⊂ PI be V -generic and jg : V → Ult(V , g ) be the associated generic embedding, thenjα ∈ V and does not depend on g for any ordinal α.

Nam Trang

Ideals and determinacy

4 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Definition (HI and DI)• DI is the statement: CH holds and there is an ω1 -dense ideal on ω1 .
• HI is the statement: CH holds and there is an ω1 -dense ideal I on ω1 such that lettingg ⊂ PI be V -generic and jg : V → Ult(V , g ) be the associated generic embedding, thenjα ∈ V and does not depend on g for any ordinal α.
The main theorem we prove is during the 2019-2020 period is.

Nam Trang

Ideals and determinacy

4 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Definition (HI and DI)• DI is the statement: CH holds and there is an ω1 -dense ideal on ω1 .
• HI is the statement: CH holds and there is an ω1 -dense ideal I on ω1 such that lettingg ⊂ PI be V -generic and jg : V → Ult(V , g ) be the associated generic embedding, thenjα ∈ V and does not depend on g for any ordinal α.
The main theorem we prove is during the 2019-2020 period is.
Theorem (Sargsyan-T.-Wilson, 2020, [AST+ 22])
Suppose T is one of the following theories.
1

The non-stationary ideal is a strong, pseudo-homogeneous ideal on Pω1 (c).

2HI.

Nam Trang

Ideals and determinacy

4 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Definition (HI and DI)• DI is the statement: CH holds and there is an ω1 -dense ideal on ω1 .
• HI is the statement: CH holds and there is an ω1 -dense ideal I on ω1 such that lettingg ⊂ PI be V -generic and jg : V → Ult(V , g ) be the associated generic embedding, thenjα ∈ V and does not depend on g for any ordinal α.
The main theorem we prove is during the 2019-2020 period is.
Theorem (Sargsyan-T.-Wilson, 2020, [AST+ 22])
Suppose T is one of the following theories.
1

The non-stationary ideal is a strong, pseudo-homogeneous ideal on Pω1 (c).

2

HI.

Then Con(T) implies Con(ADR +Θ is regular).

Nam Trang

Ideals and determinacy

4 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Theorem (Adolf-T., 2022, [AST+ 22])Con(DI) implies Con(ADR +Θ is regular).
Combining these with early forcing results of Woodin’s (see [AST+ 22] or [Wil12]), we get.

Nam Trang

Ideals and determinacy

5 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Theorem (Adolf-T., 2022, [AST+ 22])Con(DI) implies Con(ADR +Θ is regular).
Combining these with early forcing results of Woodin’s (see [AST+ 22] or [Wil12]), we get.
Theorem
The following theories are equiconsistent.
1

ADR +Θ is regular.

2

HI.

3

The non-stationary ideal is a strong, pseudo-homogeneous ideal on Pω1 (c).

4DI.

Nam Trang

Ideals and determinacy

5 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Theorem (Adolf-T., 2022, [AST+ 22])Con(DI) implies Con(ADR +Θ is regular).
Combining these with early forcing results of Woodin’s (see [AST+ 22] or [Wil12]), we get.
Theorem
The following theories are equiconsistent.
1

ADR +Θ is regular.

2

HI.

3

The non-stationary ideal is a strong, pseudo-homogeneous ideal on Pω1 (c).

4DI.

This resolves part of Woodin’s conjecture [Woo10, Problem 12].

Nam Trang

Ideals and determinacy

5 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Main theorems (cont.)

Theorem (Adolf-T., 2022, [AST+ 22])Con(DI) implies Con(ADR +Θ is regular).
Combining these with early forcing results of Woodin’s (see [AST+ 22] or [Wil12]), we get.
Theorem
The following theories are equiconsistent.
1

ADR +Θ is regular.

2

HI.

3

The non-stationary ideal is a strong, pseudo-homogeneous ideal on Pω1 (c).

4DI.

This resolves part of Woodin’s conjecture [Woo10, Problem 12].

Nam Trang

Ideals and determinacy

5 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• Partial progress on these problems has been made:• (Woodin) (2) and (3) are consistent relative to (1).
• (Steel-Zoble) MM(c) implies AD holds in L(R).

Nam Trang

Ideals and determinacy

6 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• Partial progress on these problems has been made:• (Woodin) (2) and (3) are consistent relative to (1).
• (Steel-Zoble) MM(c) implies AD holds in L(R).

• If one strengthens the hypotheses of MM(c), one can get consistency of (1). For instance,
Theorem (T., [Tra16])
If MM holds, then there is a model of ADR +Θ is regular.

Nam Trang

Ideals and determinacy

6 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• Partial progress on these problems has been made:• (Woodin) (2) and (3) are consistent relative to (1).
• (Steel-Zoble) MM(c) implies AD holds in L(R).

• If one strengthens the hypotheses of MM(c), one can get consistency of (1). For instance,
Theorem (T., [Tra16])
If MM holds, then there is a model of ADR +Θ is regular.
The other part of [Woo10, Problem 12] asks whether ADR + “Θ is regular" is equiconsistent
with MM(c)+“JNS is weakly presaturated".

Nam Trang

Ideals and determinacy

6 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• The theories “there is a precipitous ideal on ω1 " and “there is a measurable cardinal" areequiconsistent.

Nam Trang

Ideals and determinacy

7 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• The theories “there is a precipitous ideal on ω1 " and “there is a measurable cardinal" areequiconsistent.
• (Shelah, Jensen-Steel) The theories “there is a (pre)saturated ideal on ω1 " and “there is a
Woodin cardinal" are equiconsistent.

Nam Trang

Ideals and determinacy

7 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• The theories “there is a precipitous ideal on ω1 " and “there is a measurable cardinal" areequiconsistent.
• (Shelah, Jensen-Steel) The theories “there is a (pre)saturated ideal on ω1 " and “there is a
Woodin cardinal" are equiconsistent.
• (Woodin) The theories “there is a dense ideal on ω1 ", AD, “there are ω many Woodin
cardinals" are equiconsistent.

Nam Trang

Ideals and determinacy

7 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• The theories “there is a precipitous ideal on ω1 " and “there is a measurable cardinal" areequiconsistent.
• (Shelah, Jensen-Steel) The theories “there is a (pre)saturated ideal on ω1 " and “there is a
Woodin cardinal" are equiconsistent.
• (Woodin) The theories “there is a dense ideal on ω1 ", AD, “there are ω many Woodin
cardinals" are equiconsistent.
• (Ketchersid) ADR is consistent relative to “HI + ."

Nam Trang

Ideals and determinacy

7 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Some background and history• The theories “there is a precipitous ideal on ω1 " and “there is a measurable cardinal" areequiconsistent.
• (Shelah, Jensen-Steel) The theories “there is a (pre)saturated ideal on ω1 " and “there is a
Woodin cardinal" are equiconsistent.
• (Woodin) The theories “there is a dense ideal on ω1 ", AD, “there are ω many Woodin
cardinals" are equiconsistent.
• (Ketchersid) ADR is consistent relative to “HI + ."• Completing Ketchersid’s thesis was a major direction for the community for many years.
Many major meetings and work had been produced (e.g. Schindler-Steel "Core Model
Induction" manuscript, Sargsyan’s thesis, Wilson’s thesis, etc.)

Nam Trang

Ideals and determinacy

7 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

A two-player game with perfect information
Let X be a set. Suppose A ⊆ X ω . Let GA be the following game. Players I and II alternatively
play elements of X . Let xk ∈ X be the element played at the kth move.
Player
I
II

0
x0

1
x2
x1

...

n
x2n

x3

Nam Trang

...
x2n+1

Ideals and determinacy

8 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

A two-player game with perfect information
Let X be a set. Suppose A ⊆ X ω . Let GA be the following game. Players I and II alternatively
play elements of X . Let xk ∈ X be the element played at the kth move.
Player
I
II

0
x0

1
x2
x1

...

n
x2n

x3

...
x2n+1

Let
x = (xk )k<ω

Nam Trang

Ideals and determinacy

8 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

A two-player game with perfect information
Let X be a set. Suppose A ⊆ X ω . Let GA be the following game. Players I and II alternatively
play elements of X . Let xk ∈ X be the element played at the kth move.
Player
I
II

0
x0

1
x2
x1

...

n
x2n

x3

...
x2n+1

Let
x = (xk )k<ω
Then I wins the play iff x ∈ A.

Nam Trang

Ideals and determinacy

8 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

A two-player game with perfect information
Let X be a set. Suppose A ⊆ X ω . Let GA be the following game. Players I and II alternatively
play elements of X . Let xk ∈ X be the element played at the kth move.
Player
I
II

0
x0

1
x2
x1

...

n
x2n

x3

...
x2n+1

Let
x = (xk )k<ω
Then I wins the play iff x ∈ A.
I ’s strategy is a function σ : X Even → X and II ’s strategy is a function τ : X Odd → X .
GA is determined if one of the players has a winning strategy.

Nam Trang

Ideals and determinacy

8 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

A two-player game with perfect information
Let X be a set. Suppose A ⊆ X ω . Let GA be the following game. Players I and II alternatively
play elements of X . Let xk ∈ X be the element played at the kth move.
Player
I
II

0
x0

1
x2
x1

...

n
x2n

x3

...
x2n+1

Let
x = (xk )k<ω
Then I wins the play iff x ∈ A.
I ’s strategy is a function σ : X Even → X and II ’s strategy is a function τ : X Odd → X .
GA is determined if one of the players has a winning strategy.
Definition (Axiom of Determinacy, AD)
ADX is the statement: for every A ⊆ X ω , the game GA is determined.
AD is ADω .

Nam Trang

Ideals and determinacy

8 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.
• AD implies many regularity properties for sets of reals: Lebesgue measurability, Baire
property, perfect set property.

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.
• AD implies many regularity properties for sets of reals: Lebesgue measurability, Baire
property, perfect set property.
• ADR is stronger than AD.

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.
• AD implies many regularity properties for sets of reals: Lebesgue measurability, Baire
property, perfect set property.
• ADR is stronger than AD.
• AD < ADR < ADR + DC < ADR +Θ is regular < LSA.

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.
• AD implies many regularity properties for sets of reals: Lebesgue measurability, Baire
property, perfect set property.
• ADR is stronger than AD.
• AD < ADR < ADR + DC < ADR +Θ is regular < LSA.
• All of the above theories are known to be weaker than the existence of a Woodin limit ofWoodin cardinals. These theories were thought to be out of reach of inner model theory (bySolovay in the 60’s and 70’s, Woodin and others in the 80’s and 90’s).

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.
• AD implies many regularity properties for sets of reals: Lebesgue measurability, Baire
property, perfect set property.
• ADR is stronger than AD.
• AD < ADR < ADR + DC < ADR +Θ is regular < LSA.
• All of the above theories are known to be weaker than the existence of a Woodin limit ofWoodin cardinals. These theories were thought to be out of reach of inner model theory (bySolovay in the 60’s and 70’s, Woodin and others in the 80’s and 90’s).
• ADω1 , ADP(R) are “too strong".

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Axioms of Determinacy

Definition (Axiom of Determinacy, AD:)ADX is the statement for every A ⊆ X ω , the game GA is determined.
AD is ADω .
• Let Θ be the supremum of α such that there is a surjection of R onto α.
• AD implies many regularity properties for sets of reals: Lebesgue measurability, Baire
property, perfect set property.
• ADR is stronger than AD.
• AD < ADR < ADR + DC < ADR +Θ is regular < LSA.
• All of the above theories are known to be weaker than the existence of a Woodin limit ofWoodin cardinals. These theories were thought to be out of reach of inner model theory (bySolovay in the 60’s and 70’s, Woodin and others in the 80’s and 90’s).
• ADω1 , ADP(R) are “too strong".

Nam Trang

Ideals and determinacy

9 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Unconventional methods

Conjecture: (Woodin, 90’s) ADR + “Θ is regular” is very strong, beyond the reach of all large
cardinals and forcing axioms. So is LSA.

Nam Trang

Ideals and determinacy

10 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Unconventional methods

Conjecture: (Woodin, 90’s) ADR + “Θ is regular” is very strong, beyond the reach of all large
cardinals and forcing axioms. So is LSA.
There have been debates over whether ADR + “Θ is regular” is a reasonable axiom or not.

Nam Trang

Ideals and determinacy

10 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Unconventional methods

Conjecture: (Woodin, 90’s) ADR + “Θ is regular” is very strong, beyond the reach of all large
cardinals and forcing axioms. So is LSA.
There have been debates over whether ADR + “Θ is regular” is a reasonable axiom or not.
Theorem (Woodin, early 2000’s, Sargsyan, 2008, [Sar15])
Assume the existence of large cardinals. Then there is a universe in which ADR + “Θ is regular”
holds.

Nam Trang

Ideals and determinacy

10 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Unconventional methods

Conjecture: (Woodin, 90’s) ADR + “Θ is regular” is very strong, beyond the reach of all large
cardinals and forcing axioms. So is LSA.
There have been debates over whether ADR + “Θ is regular” is a reasonable axiom or not.
Theorem (Woodin, early 2000’s, Sargsyan, 2008, [Sar15])
Assume the existence of large cardinals. Then there is a universe in which ADR + “Θ is regular”
holds.
Theorem ([ST])Con(WLW) implies Con(LSA).

Nam Trang

Ideals and determinacy

10 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Unconventional methods

Conjecture: (Woodin, 90’s) ADR + “Θ is regular” is very strong, beyond the reach of all large
cardinals and forcing axioms. So is LSA.
There have been debates over whether ADR + “Θ is regular” is a reasonable axiom or not.
Theorem (Woodin, early 2000’s, Sargsyan, 2008, [Sar15])
Assume the existence of large cardinals. Then there is a universe in which ADR + “Θ is regular”
holds.
Theorem ([ST])Con(WLW) implies Con(LSA).

Nam Trang

Ideals and determinacy

10 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Hierarchy relations
Large Cardinals

Determinacy

HOD

Supercompact

Combinatorial Theories

PFA, MM

WLW

WLW

ω1 is (str/super)compact
Sealing?

LSA − over − UB

excellent?

LSA

lsa-type hod pairΘ reg-hypo

ADR +Θ regular

Regular lim of Wdns

ω1 is P(R)-spct, MM(c)non-domestic

ADR + DC

ω1 Woodins

ω1 is P(R)-str.cpct.

ADR -hypo

ADRω Woodinsω WoodinsAD

1 Woodin1 measurable

Analytic Determinacy

ZFC

Borel Determinacy

Nam Trang

AD + ω1 is R-str.cpct.
Precipitious ideal on ω1

Ideals and determinacy

11 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Universally Baire sets

Definition (Feng-Magidor-Woodin)
A ⊆ R is κ-universally Baire if there are trees T , U ⊆ (ω × ON)<ω that are κ-absolutely
complemented, i.e. A = p[T ] = R\p[U] and whenever P is a forcing such that |P| < κ and g ⊆ P
is V -generic, in V [g ], p[T ] = R\p[U]. In this case, we let Ag = p[T ] be the canonicalinterpretation of A in V [g ].
A is universally Baire if A is κ-universally Baire for all κ. Let Γ∞ be the collection of all
universally Baire sets.

Nam Trang

Ideals and determinacy

12 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Universally Baire sets

Definition (Feng-Magidor-Woodin)
A ⊆ R is κ-universally Baire if there are trees T , U ⊆ (ω × ON)<ω that are κ-absolutely
complemented, i.e. A = p[T ] = R\p[U] and whenever P is a forcing such that |P| < κ and g ⊆ P
is V -generic, in V [g ], p[T ] = R\p[U]. In this case, we let Ag = p[T ] be the canonicalinterpretation of A in V [g ].
A is universally Baire if A is κ-universally Baire for all κ. Let Γ∞ be the collection of all
universally Baire sets.
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.

Nam Trang

Ideals and determinacy

12 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Universally Baire sets

Definition (Feng-Magidor-Woodin)
A ⊆ R is κ-universally Baire if there are trees T , U ⊆ (ω × ON)<ω that are κ-absolutely
complemented, i.e. A = p[T ] = R\p[U] and whenever P is a forcing such that |P| < κ and g ⊆ P
is V -generic, in V [g ], p[T ] = R\p[U]. In this case, we let Ag = p[T ] be the canonicalinterpretation of A in V [g ].
A is universally Baire if A is κ-universally Baire for all κ. Let Γ∞ be the collection of all
universally Baire sets.
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?

Nam Trang

Ideals and determinacy

12 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Universally Baire sets

Definition (Feng-Magidor-Woodin)
A ⊆ R is κ-universally Baire if there are trees T , U ⊆ (ω × ON)<ω that are κ-absolutely
complemented, i.e. A = p[T ] = R\p[U] and whenever P is a forcing such that |P| < κ and g ⊆ P
is V -generic, in V [g ], p[T ] = R\p[U]. In this case, we let Ag = p[T ] be the canonicalinterpretation of A in V [g ].
A is universally Baire if A is κ-universally Baire for all κ. Let Γ∞ be the collection of all
universally Baire sets.
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?

Nam Trang

Ideals and determinacy

12 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?
• Is there a Γ ⊆ Γ∞ such that the HOD of L(Γ, R) satisfies large large cardinals?

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?
• Is there a Γ ⊆ Γ∞ such that the HOD of L(Γ, R) satisfies large large cardinals?
• Is there A ⊂ R such that A ∈/ Γ∞ such that L(A, R)  AD+ ?

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?
• Is there a Γ ⊆ Γ∞ such that the HOD of L(Γ, R) satisfies large large cardinals?
• Is there A ⊂ R such that A ∈/ Γ∞ such that L(A, R)  AD+ ?
Under the hypothesis above, the core model induction shows that there are complicated models
of determinacy within Γ∞ , e.g. there are Γ ⊂ Γ∞ such that L(Γ, R)  ADR +Θ is regular.

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?
• Is there a Γ ⊆ Γ∞ such that the HOD of L(Γ, R) satisfies large large cardinals?
• Is there A ⊂ R such that A ∈/ Γ∞ such that L(A, R)  AD+ ?
Under the hypothesis above, the core model induction shows that there are complicated models
of determinacy within Γ∞ , e.g. there are Γ ⊂ Γ∞ such that L(Γ, R)  ADR +Θ is regular.
Remark:• Divergent models of AD+ sit at the limit of our ability to construct iterable hod mice andhence is an obstacle for current techniques of pushing up lowerbound consistency strength.

Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?
• Is there a Γ ⊆ Γ∞ such that the HOD of L(Γ, R) satisfies large large cardinals?
• Is there A ⊂ R such that A ∈/ Γ∞ such that L(A, R)  AD+ ?
Under the hypothesis above, the core model induction shows that there are complicated models
of determinacy within Γ∞ , e.g. there are Γ ⊂ Γ∞ such that L(Γ, R)  ADR +Θ is regular.
Remark:• Divergent models of AD+ sit at the limit of our ability to construct iterable hod mice andhence is an obstacle for current techniques of pushing up lowerbound consistency strength.
• Sealing of Γ∞ turns out to be the next major obstacle for the CMI, [ST19].
Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Martin’s Maximum and universally Baire sets
Assume MM++ and there is a proper class of Woodin cardinals. The following question is ofgreat interest.
Question
How complicated is Γ∞ ?
Specific interpretations of the question:• What is the rank of the Wadge prewellordering of Γ∞ ?
• Is there a Γ ⊆ Γ∞ such that L(Γ, R) satisfies a strong form of determinacy, e.g. ADR +Θ isregular, LSA etc?
• Is there a Γ ⊆ Γ∞ such that the HOD of L(Γ, R) satisfies large large cardinals?
• Is there A ⊂ R such that A ∈/ Γ∞ such that L(A, R)  AD+ ?
Under the hypothesis above, the core model induction shows that there are complicated models
of determinacy within Γ∞ , e.g. there are Γ ⊂ Γ∞ such that L(Γ, R)  ADR +Θ is regular.
Remark:• Divergent models of AD+ sit at the limit of our ability to construct iterable hod mice andhence is an obstacle for current techniques of pushing up lowerbound consistency strength.
• Sealing of Γ∞ turns out to be the next major obstacle for the CMI, [ST19].
Nam Trang

Ideals and determinacy

13 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction• Via core model theory, one has that PFA implies there are models of ADR + DC. This is atheorem of Jensen-Schimmerling-Schindler-Steel [JSSS09].

Nam Trang

Ideals and determinacy

14 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction• Via core model theory, one has that PFA implies there are models of ADR + DC. This is atheorem of Jensen-Schimmerling-Schindler-Steel [JSSS09].
• The method above does not seem to generalize, e.g. it does not give models of ADR +Θ isregular. Generalizing the Andretta-Neeman-Steel result is highly nontrivial.

Nam Trang

Ideals and determinacy

14 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction• Via core model theory, one has that PFA implies there are models of ADR + DC. This is atheorem of Jensen-Schimmerling-Schindler-Steel [JSSS09].
• The method above does not seem to generalize, e.g. it does not give models of ADR +Θ isregular. Generalizing the Andretta-Neeman-Steel result is highly nontrivial.
• CMI does better in this respect.

Nam Trang

Ideals and determinacy

14 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction• Via core model theory, one has that PFA implies there are models of ADR + DC. This is atheorem of Jensen-Schimmerling-Schindler-Steel [JSSS09].
• The method above does not seem to generalize, e.g. it does not give models of ADR +Θ isregular. Generalizing the Andretta-Neeman-Steel result is highly nontrivial.
• CMI does better in this respect.

Nam Trang

Ideals and determinacy

14 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction (cont.)• The core model induction is a general method, pioneered by Woodin, further developed by
Steel, Schindler, and others, used for mining strength of a given theory. In a typical coremodel induction, one constructs models of determinacy (inductively) that extend one anotherand this is achieved by constructing mouse operators that capture the relevant sets of reals.

Nam Trang

Ideals and determinacy

15 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction (cont.)• The core model induction is a general method, pioneered by Woodin, further developed by
Steel, Schindler, and others, used for mining strength of a given theory. In a typical coremodel induction, one constructs models of determinacy (inductively) that extend one anotherand this is achieved by constructing mouse operators that capture the relevant sets of reals.
• Examples of mouse operators are: x 7→ x ] , x 7→ M]1 (x), a pair (P, Σ) where P is a finestructural mouse (or a hod mouse) and Σ is P’s iteration strategy with nice condensation
property (roughly, trees according to Σ collapses to trees according Σ).

Nam Trang

Ideals and determinacy

15 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction (cont.)• The core model induction is a general method, pioneered by Woodin, further developed by
Steel, Schindler, and others, used for mining strength of a given theory. In a typical coremodel induction, one constructs models of determinacy (inductively) that extend one anotherand this is achieved by constructing mouse operators that capture the relevant sets of reals.
• Examples of mouse operators are: x 7→ x ] , x 7→ M]1 (x), a pair (P, Σ) where P is a finestructural mouse (or a hod mouse) and Σ is P’s iteration strategy with nice condensation
property (roughly, trees according to Σ collapses to trees according Σ).
• This relationship is given by a theorem of Neeman: if N is a (sufficiently iterable) mousewith iteration strategy Σ and N  δ is Woodin. Suppose N captures A ⊆ R, i.e. there is aCol(ω, δ)-term relation τA in N such that whenever i : N → M is according to Σ, lettingg ⊆ Col(ω, i(δ)) be M-generic, then A ∩ M[g ] = (i(τA )g ). Then A is determined.

Nam Trang

Ideals and determinacy

15 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Core model induction (cont.)• The core model induction is a general method, pioneered by Woodin, further developed by
Steel, Schindler, and others, used for mining strength of a given theory. In a typical coremodel induction, one constructs models of determinacy (inductively) that extend one anotherand this is achieved by constructing mouse operators that capture the relevant sets of reals.
• Examples of mouse operators are: x 7→ x ] , x 7→ M]1 (x), a pair (P, Σ) where P is a finestructural mouse (or a hod mouse) and Σ is P’s iteration strategy with nice condensation
property (roughly, trees according to Σ collapses to trees according Σ).
• This relationship is given by a theorem of Neeman: if N is a (sufficiently iterable) mousewith iteration strategy Σ and N  δ is Woodin. Suppose N captures A ⊆ R, i.e. there is aCol(ω, δ)-term relation τA in N such that whenever i : N → M is according to Σ, lettingg ⊆ Col(ω, i(δ)) be M-generic, then A ∩ M[g ] = (i(τA )g ). Then A is determined.
• Often times (as it is the case of the CMI here), one shows that the operators constructed arecoded by sets of reals that are κ-UB for some κ. For example, in the Adolf-T. theorem, weshow the operators are ω2 -UB.

Nam Trang

Ideals and determinacy

15 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .
• Let P be either Coll(ω1 , R) or Pmax . (We can abstract the essential properties of these
forcings that we need). In particular, P is countably closed, homogeneous, and P ⊂ R.

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .
• Let P be either Coll(ω1 , R) or Pmax . (We can abstract the essential properties of these
forcings that we need). In particular, P is countably closed, homogeneous, and P ⊂ R.
• Let G ⊆ P be V -generic. Can show that V [G ] satisfies c-DC and Θ = c+ . (Steel-Van
Wesep, Woodin’s Pmax -analysis)

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .
• Let P be either Coll(ω1 , R) or Pmax . (We can abstract the essential properties of these
forcings that we need). In particular, P is countably closed, homogeneous, and P ⊂ R.
• Let G ⊆ P be V -generic. Can show that V [G ] satisfies c-DC and Θ = c+ . (Steel-Van
Wesep, Woodin’s Pmax -analysis)• Let Q be Coll(Θ, P(R)V ) and H ⊆ Q be V [G ]-generic. Q does not add subsets of Pω1 (R).

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .
• Let P be either Coll(ω1 , R) or Pmax . (We can abstract the essential properties of these
forcings that we need). In particular, P is countably closed, homogeneous, and P ⊂ R.
• Let G ⊆ P be V -generic. Can show that V [G ] satisfies c-DC and Θ = c+ . (Steel-Van
Wesep, Woodin’s Pmax -analysis)• Let Q be Coll(Θ, P(R)V ) and H ⊆ Q be V [G ]-generic. Q does not add subsets of Pω1 (R).
• Note that V [G ][H] satisfies AC and RV = RV [G ][H] .

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .
• Let P be either Coll(ω1 , R) or Pmax . (We can abstract the essential properties of these
forcings that we need). In particular, P is countably closed, homogeneous, and P ⊂ R.
• Let G ⊆ P be V -generic. Can show that V [G ] satisfies c-DC and Θ = c+ . (Steel-Van
Wesep, Woodin’s Pmax -analysis)• Let Q be Coll(Θ, P(R)V ) and H ⊆ Q be V [G ]-generic. Q does not add subsets of Pω1 (R).
• Note that V [G ][H] satisfies AC and RV = RV [G ][H] .
• We in fact will show in V [G ][H] there is an ideal I with the ordinal covering property withrespect to V , i.e. for every function F : Pω1 (R) → ON and every I-positive set S, there issome I-positive set S0 ⊆ S and some F0 : Pω1 (R) → ON in V such that F S0 = F0 S0 .

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Set up• Assume V satisfies V = L(P(R)) + ADR +Θ is regular. We show there is a strong,pseudo-homogeneous ideal on Pω1 (R) in some generic extension of V .
• Let P be either Coll(ω1 , R) or Pmax . (We can abstract the essential properties of these
forcings that we need). In particular, P is countably closed, homogeneous, and P ⊂ R.
• Let G ⊆ P be V -generic. Can show that V [G ] satisfies c-DC and Θ = c+ . (Steel-Van
Wesep, Woodin’s Pmax -analysis)• Let Q be Coll(Θ, P(R)V ) and H ⊆ Q be V [G ]-generic. Q does not add subsets of Pω1 (R).
• Note that V [G ][H] satisfies AC and RV = RV [G ][H] .
• We in fact will show in V [G ][H] there is an ideal I with the ordinal covering property withrespect to V , i.e. for every function F : Pω1 (R) → ON and every I-positive set S, there issome I-positive set S0 ⊆ S and some F0 : Pω1 (R) → ON in V such that F S0 = F0 S0 .

Nam Trang

Ideals and determinacy

16 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Ordinal covering implies strong, pseudo-homogeneity
Lemma
Suppose V , G , H are as above. Suppose I is an ideal on Pω1 (R) with the ordinal covering
property. Let µ be the Solovay measure. Let K ⊂ PI be V [G ][H]-generic, then(a)the generic embedding jK ON = jµ ON.

(b)

I is strong.

Nam Trang

Ideals and determinacy

17 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Ordinal covering implies strong, pseudo-homogeneity
Lemma
Suppose V , G , H are as above. Suppose I is an ideal on Pω1 (R) with the ordinal covering
property. Let µ be the Solovay measure. Let K ⊂ PI be V [G ][H]-generic, then(a)the generic embedding jK ON = jµ ON.

(b)

I is strong.

Proof.
For (a), for any F : Pω1 (R) → ON in V [G ][H], using the covering property, it is easy to see that
there is some S ∈ K and F0 ∈ V such that F S = F0 S. Also, K ∩ V = µ, so this gives{F : Pω1 (R) → ON}V [G ][H] /K = {F : Pω1 (R) → ON}V /µand jK ON = jµ ON. Part (b) follows from (a) and the fact that jµ (ω1 ) = ΘV = c+ .

Nam Trang

Ideals and determinacy

17 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Ordinal covering implies strong, pseudo-homogeneity
Lemma
Suppose V , G , H are as above. Suppose I is an ideal on Pω1 (R) with the ordinal covering
property. Let µ be the Solovay measure. Let K ⊂ PI be V [G ][H]-generic, then(a)the generic embedding jK ON = jµ ON.

(b)

I is strong.

Proof.
For (a), for any F : Pω1 (R) → ON in V [G ][H], using the covering property, it is easy to see that
there is some S ∈ K and F0 ∈ V such that F S = F0 S. Also, K ∩ V = µ, so this gives{F : Pω1 (R) → ON}V [G ][H] /K = {F : Pω1 (R) → ON}V /µand jK ON = jµ ON. Part (b) follows from (a) and the fact that jµ (ω1 ) = ΘV = c+ .
Lemma
In V [G ][H], if I has the ordinal covering property relative to V , then I is pseudo-homogeneous.

Nam Trang

Ideals and determinacy

17 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Ordinal covering implies strong, pseudo-homogeneity
Lemma
Suppose V , G , H are as above. Suppose I is an ideal on Pω1 (R) with the ordinal covering
property. Let µ be the Solovay measure. Let K ⊂ PI be V [G ][H]-generic, then(a)the generic embedding jK ON = jµ ON.

(b)

I is strong.

Proof.
For (a), for any F : Pω1 (R) → ON in V [G ][H], using the covering property, it is easy to see that
there is some S ∈ K and F0 ∈ V such that F S = F0 S. Also, K ∩ V = µ, so this gives{F : Pω1 (R) → ON}V [G ][H] /K = {F : Pω1 (R) → ON}V /µand jK ON = jµ ON. Part (b) follows from (a) and the fact that jµ (ω1 ) = ΘV = c+ .
Lemma
In V [G ][H], if I has the ordinal covering property relative to V , then I is pseudo-homogeneous.

Nam Trang

Ideals and determinacy

17 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)NSω1 ,R has the covering property

Let I = NSω1 ,R be the nonstationary ideal on Pω1 (R) in V [G ][H].

Nam Trang

Ideals and determinacy

18 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)NSω1 ,R has the covering property

Let I = NSω1 ,R be the nonstationary ideal on Pω1 (R) in V [G ][H].
Theorem
In V [G ][H], I has the ordinal covering property with respect to V .

Nam Trang

Ideals and determinacy

18 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)NSω1 ,R has the covering property

Let I = NSω1 ,R be the nonstationary ideal on Pω1 (R) in V [G ][H].
Theorem
In V [G ][H], I has the ordinal covering property with respect to V .
Remark. A similar analysis (take e.g. P = Coll(ω1 , R)) gives HI holds in V [G ][H] (in fact, one
can show there is an ω1 -dense ideal on Pω1 (R) with the ordinal covering property, so DI holds).

Nam Trang

Ideals and determinacy

18 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)NSω1 ,R has the covering property

Let I = NSω1 ,R be the nonstationary ideal on Pω1 (R) in V [G ][H].
Theorem
In V [G ][H], I has the ordinal covering property with respect to V .
Remark. A similar analysis (take e.g. P = Coll(ω1 , R)) gives HI holds in V [G ][H] (in fact, one
can show there is an ω1 -dense ideal on Pω1 (R) with the ordinal covering property, so DI holds).

Nam Trang

Ideals and determinacy

18 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.
• To go further, the best way is to organize our induction according to the pattern of scales.
The main, nontrivial step is as follows. Suppose we have a pointclass Γ such that Γ is inductive-like, i.e. Γ has the scales property, is closed under real quantifications (i.e. Γ canbe (Σ21 )L(R) or Γ can be P(R)M where M  AD+ ).
e

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.
• To go further, the best way is to organize our induction according to the pattern of scales.
The main, nontrivial step is as follows. Suppose we have a pointclass Γ such that Γ is inductive-like, i.e. Γ has the scales property, is closed under real quantifications (i.e. Γ canbe (Σ21 )L(R) or Γ can be P(R)M where M  AD+ ).
 Γ is e
determined.

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.
• To go further, the best way is to organize our induction according to the pattern of scales.
The main, nontrivial step is as follows. Suppose we have a pointclass Γ such that Γ is inductive-like, i.e. Γ has the scales property, is closed under real quantifications (i.e. Γ canbe (Σ21 )L(R) or Γ can be P(R)M where M  AD+ ).
 Γ is e
determined.
 Γ-SMC holds.

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.
• To go further, the best way is to organize our induction according to the pattern of scales.
The main, nontrivial step is as follows. Suppose we have a pointclass Γ such that Γ is inductive-like, i.e. Γ has the scales property, is closed under real quantifications (i.e. Γ canbe (Σ21 )L(R) or Γ can be P(R)M where M  AD+ ).
 Γ is e
determined.
 Γ-SMC holds.

The goal in this case is to construct a hod pair (P, Σ) that “transcends" Γ. In particular, we
want
 Σ∈
/ Γ,
 Σ is self-scaled, i.e. all projective sets in Σ are determined and have projective-in-Σ scales, and

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.
• To go further, the best way is to organize our induction according to the pattern of scales.
The main, nontrivial step is as follows. Suppose we have a pointclass Γ such that Γ is inductive-like, i.e. Γ has the scales property, is closed under real quantifications (i.e. Γ canbe (Σ21 )L(R) or Γ can be P(R)M where M  AD+ ).
 Γ is e
determined.
 Γ-SMC holds.

The goal in this case is to construct a hod pair (P, Σ) that “transcends" Γ. In particular, we
want
 Σ∈
/ Γ,
 Σ is self-scaled, i.e. all projective sets in Σ are determined and have projective-in-Σ scales, and every A ∈ Γ is Wadge reducible to Σ.

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence• General successor step. Given a nice operator J, use the K J -dichotomy to show that foreach x ∈ dom(J), M1J,] (x) exists.
• To go further, the best way is to organize our induction according to the pattern of scales.
The main, nontrivial step is as follows. Suppose we have a pointclass Γ such that Γ is inductive-like, i.e. Γ has the scales property, is closed under real quantifications (i.e. Γ canbe (Σ21 )L(R) or Γ can be P(R)M where M  AD+ ).
 Γ is e
determined.
 Γ-SMC holds.

The goal in this case is to construct a hod pair (P, Σ) that “transcends" Γ. In particular, we
want
 Σ∈
/ Γ,
 Σ is self-scaled, i.e. all projective sets in Σ are determined and have projective-in-Σ scales, and every A ∈ Γ is Wadge reducible to Σ.

Nam Trang

Ideals and determinacy

19 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1

Nam Trang

Lp Σ (R)or more generally, Ω = Σ1

Ideals and determinacy.

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

2

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.
~ = hAi : i < ωi ∈ Env (Ω)ω such that A0 is a universal
Get a self-justifying system (sjs) A^and scales.
Ω̌-set. A sjs is a system that is closed under complements

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

2

3

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.
~ = hAi : i < ωi ∈ Env (Ω)ω such that A0 is a universal
Get a self-justifying system (sjs) A^and scales.
Ω̌-set. A sjs is a system that is closed under complements~
Show there is an Ω-suitable (hybrid) premouse P with an A-guided
iteration strategy Λ.

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

2

3
4

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.
~ = hAi : i < ωi ∈ Env (Ω)ω such that A0 is a universal
Get a self-justifying system (sjs) A^and scales.
Ω̌-set. A sjs is a system that is closed under complements~
Show there is an Ω-suitable (hybrid) premouse P with an A-guided
iteration strategy Λ.
Show Λ condenses well (e.g. show Λ has branch condensation). This follows fromcondensation properties of sjs.

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

2

3
4

5

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.
~ = hAi : i < ωi ∈ Env (Ω)ω such that A0 is a universal
Get a self-justifying system (sjs) A^and scales.
Ω̌-set. A sjs is a system that is closed under complements~
Show there is an Ω-suitable (hybrid) premouse P with an A-guided
iteration strategy Λ.
Show Λ condenses well (e.g. show Λ has branch condensation). This follows fromcondensation properties of sjs.
Thus, we can make sense of LpΛ (R) and show that LpΛ (R)  AD+ .

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

2

3
4

5
6

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.
~ = hAi : i < ωi ∈ Env (Ω)ω such that A0 is a universal
Get a self-justifying system (sjs) A^and scales.
Ω̌-set. A sjs is a system that is closed under complements~
Show there is an Ω-suitable (hybrid) premouse P with an A-guided
iteration strategy Λ.
Show Λ condenses well (e.g. show Λ has branch condensation). This follows fromcondensation properties of sjs.
Thus, we can make sense of LpΛ (R) and show that LpΛ (R)  AD+ .
This gives us that Λ ∈ Γ\Env (Ω) as desired.
^

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Successor stages in the Solovay sequence (cont.)

Let Ω be an inductive-like, nonselfdual pointclass such that Ω is determined, has the scales
L(R)property, and Γ-SMC holds. Most important examples, Ω = Σ1
1

2

3
4

5
6

Lp Σ (R)or more generally, Ω = Σ1.

Construct a scale on a universal Ω̌-set whose norm relations are in the boldface envelope
Env (Ω). Here A ∈ Env (Ω) iff there is a real x, for each countable σ ⊂ R, there is^B^∈ Ω(x) ∩ Ω̌(x) suchthat A ∩ σ = B ∩ σ.
~ = hAi : i < ωi ∈ Env (Ω)ω such that A0 is a universal
Get a self-justifying system (sjs) A^and scales.
Ω̌-set. A sjs is a system that is closed under complements~
Show there is an Ω-suitable (hybrid) premouse P with an A-guided
iteration strategy Λ.
Show Λ condenses well (e.g. show Λ has branch condensation). This follows fromcondensation properties of sjs.
Thus, we can make sense of LpΛ (R) and show that LpΛ (R)  AD+ .
This gives us that Λ ∈ Γ\Env (Ω) as desired.
^

Nam Trang

Ideals and determinacy

20 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages in the Solovay sequence
Suppose Ω is inductive-like, determined, has the scales property, constructibly closed, and Γ-SMC
holds. Important examples, Ω = P(R)M where M  AD+ + ADR .

Nam Trang

Ideals and determinacy

21 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages in the Solovay sequence
Suppose Ω is inductive-like, determined, has the scales property, constructibly closed, and Γ-SMC
holds. Important examples, Ω = P(R)M where M  AD+ + ADR .
1

Let H be the hod limit of M. Let λ be the length of the Solovay sequence of M and for eachα < λ, let Σα be the strategy of the α-th layer of H, H(α). Σα is in Ult(V , G ) but by
pseudo-homogeneity, ΣV ∈ V .

Nam Trang

Ideals and determinacy

21 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages in the Solovay sequence
Suppose Ω is inductive-like, determined, has the scales property, constructibly closed, and Γ-SMC
holds. Important examples, Ω = P(R)M where M  AD+ + ADR .
1

2

Let H be the hod limit of M. Let λ be the length of the Solovay sequence of M and for eachα < λ, let Σα be the strategy of the α-th layer of H, H(α). Σα is in Ult(V , G ) but by
pseudo-homogeneity, ΣV ∈ V .
Let H+ = Lp ⊕α Σα (H) and let Λ be a j-realizable strategy of H+ extending ⊕α Σα . Let++δ H be the sup of the Woodin cardinals of H+ , so δ H = o(H).

Nam Trang

Ideals and determinacy

21 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages in the Solovay sequence
Suppose Ω is inductive-like, determined, has the scales property, constructibly closed, and Γ-SMC
holds. Important examples, Ω = P(R)M where M  AD+ + ADR .
1

2

3

Let H be the hod limit of M. Let λ be the length of the Solovay sequence of M and for eachα < λ, let Σα be the strategy of the α-th layer of H, H(α). Σα is in Ult(V , G ) but by
pseudo-homogeneity, ΣV ∈ V .
Let H+ = Lp ⊕α Σα (H) and let Λ be a j-realizable strategy of H+ extending ⊕α Σα . Let++δ H be the sup of the Woodin cardinals of H+ , so δ H = o(H).
Show that some Λ is j(Ω)-fullness preserving and has j-condensation. Roughly, this ensures~ is a non-dropping stack according to Λ in Ult(V , G ) withthat the following hold: suppose T
last model R, then the realization map σ : R → j(H+ ) defined as~Ψσ(π T (f )(a)) = j(f )(πR|δR ,∞ (a)),

where f ∈ H+ , a ∈ [δ R ]<ω , Ψ is the pullback strategy of j(H) has the property thatΨσδ R = πR|δR ,∞ .

Nam Trang

Ideals and determinacy

21 / 26

Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages in the Solovay sequence
Suppose Ω is inductive-like, determined, has the scales property, constructibly closed, and Γ-SMC
holds. Important examples, Ω = P(R)M where M  AD+ + ADR .
1

2

3

Let H be the hod limit of M. Let λ be the length of the Solovay sequence of M and for eachα < λ, let Σα be the strategy of the α-th layer of H, H(α). Σα is in Ult(V , G ) but by
pseudo-homogeneity, ΣV ∈ V .
Let H+ = Lp ⊕α Σα (H) and let Λ be a j-realizable strategy of H+ extending ⊕α Σα . Let++δ H be the sup of the Woodin cardinals of H+ , so δ H = o(H).
Show that some Λ is j(Ω)-fullness preserving and has j-condensation. Roughly, this ensures~ is a non-dropping stack according to Λ in Ult(V , G ) withthat the following hold: suppose T
last model R, then the realization map σ : R → j(H+ ) defined as~Ψσ(π T (f )(a)) = j(f )(πR|δR ,∞ (a)),

where f ∈ H+ , a ∈ [δ R ]<ω , Ψ is the pullback strategy of j(H) has the property thatΨσδ R = πR|δR ,∞ .
4

Let M∞ = M∞ (H+ , Λ). Let σ : M∞ → j(H+ ) be the limit of the maps described above.
ΛThen jH+ = σ ◦ πH
and crt(σ) = δ M∞ . This implies L(j(Γ)|δ M∞ )  ADR +Θ is+ ,M∞regular. We’re done.

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages and strategies with strong hull condensation
Definition (Steel, [Ste22])
An iteration strategy Σ of a hod mouse P has strong hull condensation if whenever U is a
pseudo-hull of a stack T according to Σ, then U is according to Σ.

Nam Trang

Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages and strategies with strong hull condensation
Definition (Steel, [Ste22])
An iteration strategy Σ of a hod mouse P has strong hull condensation if whenever U is a
pseudo-hull of a stack T according to Σ, then U is according to Σ.
Lemma (Adolf-T.)
If (P, Σ) is a hod pair such that Σ has branch condensation and is fullness preserving, then Σ hasstrong hull condensation.

Nam Trang

Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages and strategies with strong hull condensation
Definition (Steel, [Ste22])
An iteration strategy Σ of a hod mouse P has strong hull condensation if whenever U is a
pseudo-hull of a stack T according to Σ, then U is according to Σ.
Lemma (Adolf-T.)
If (P, Σ) is a hod pair such that Σ has branch condensation and is fullness preserving, then Σ hasstrong hull condensation.
Lemma (Adolf-T.)
Assume CH+ there is an ω1 -dense ideal I on ω1 . Suppose (R, Λ) is a hod pair such that
|R|V ≤ ω1 and Λ is an ω2 -iteration strategy for R with strong hull condensation and is fullness
preserving. Let g ⊆ PI be V -generic and j = jg : V → M be the corresponding genericembedding. Then Λ = j(Λ)j .

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Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Limit stages and strategies with strong hull condensation
Definition (Steel, [Ste22])
An iteration strategy Σ of a hod mouse P has strong hull condensation if whenever U is a
pseudo-hull of a stack T according to Σ, then U is according to Σ.
Lemma (Adolf-T.)
If (P, Σ) is a hod pair such that Σ has branch condensation and is fullness preserving, then Σ hasstrong hull condensation.
Lemma (Adolf-T.)
Assume CH+ there is an ω1 -dense ideal I on ω1 . Suppose (R, Λ) is a hod pair such that
|R|V ≤ ω1 and Λ is an ω2 -iteration strategy for R with strong hull condensation and is fullness
preserving. Let g ⊆ PI be V -generic and j = jg : V → M be the corresponding genericembedding. Then Λ = j(Λ)j .
The proof uses substantially work of F. Schlutzenberg. The lemma allows us to show some such
hod pair is ω2 -UB and continues the core model induction.
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Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Questions and conjectures

Conjecture
The following are equiconsistent.
1

ADR +Θ is regular.

2MM(c)+“JNS is weakly presaturated. "3MM(c).

Nam Trang

Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Questions and conjectures

Conjecture
The following are equiconsistent.
1

ADR +Θ is regular.

2MM(c)+“JNS is weakly presaturated. "3MM(c).

Question
How strong is the theory “there is an ω2 -dense ideal on ω2 "?

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Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

References I
Dominik Adolf, Grigor Sargsyan, Nam Trang, Trevor Wilson, and Martin Zeman, Ideals andstrong axioms of determinacy, arXiv preprint arXiv:2111.06220 (2022), accepted to JAMS.
Ronald Jensen, Ernest Schimmerling, Ralf Schindler, and John Steel, Stacking mice, The
Journal of Symbolic Logic 74 (2009), no. 01, 315–335.
G. Sargsyan, Hod mice and the mouse set conjecture, vol. 236, Memoirs of the American
Mathematical Society, no. 1111, American Mathematical Society, 2015.
Grigor Sargsyan and Nam Trang, The largest suslin axiom, Lecture Notes in Logic,
Cambridge University Press, to appear.
, The exact consistency strength of the generic absoluteness for the universally bairesets, To appear.
John R Steel, A comparison process for mouse pairs, vol. 51, Cambridge University Press,
2022.
N. Trang, PFA and guessing models, Israel Journal of Mathematics 215 (2016), 607–667,
Available at www.math.unt.edu/∼ntrang/.
Trevor Miles Wilson, Contributions to descriptive inner model theory, Ph.D. thesis,
University of California, 2012, Available at author’s website.
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Ideals and determinacy

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

References II

W. Hugh Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal,
revised ed., De Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter GmbH& Co. KG, Berlin, 2010. MR 2723878

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Introduction
Axioms of Determinacy
CMI, Universally Baire sets, and strong forcing axioms
Forcing over models of determinacy
Core model induction (brief outline)

Thank you!

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