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Page 1: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy and Large Cardinals

Itay Neeman

Department of Mathematics

University of California Los Angeles

Los Angeles, CA 90095-1555

25 August 2006

Use PageDown or the down arrow to scroll through slides.

Press Esc when done.

0

Page 2: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

1

Page 3: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

1

Page 4: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω.

1

Page 5: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

1

Page 6: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8 · · ·II a1 a3 a5 a7 a9 · · ·

Players I and II alternate playing numbers an ∈ ω,forming tog-

ether an infinite sequence z = 〈a0, a1, a2, · · · · · · 〉 ∈ ωω.

If z belongs to A then player I wins.

If z does not belong to A then player II wins.

1

Page 7: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I

II

1

Page 8: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I

II

Players I and II alternate playing numbers an ∈ ω,

1

Page 9: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0

II

Players I and II alternate playing numbers an ∈ ω,

1

Page 10: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0

II a1

Players I and II alternate playing numbers an ∈ ω,

1

Page 11: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2

II a1

Players I and II alternate playing numbers an ∈ ω,

1

Page 12: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2

II a1 a3

Players I and II alternate playing numbers an ∈ ω,

1

Page 13: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4

II a1 a3

Players I and II alternate playing numbers an ∈ ω,

1

Page 14: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4

II a1 a3 a5

Players I and II alternate playing numbers an ∈ ω,

1

Page 15: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6

II a1 a3 a5

Players I and II alternate playing numbers an ∈ ω,

1

Page 16: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6

II a1 a3 a5 a7

Players I and II alternate playing numbers an ∈ ω,

1

Page 17: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8

II a1 a3 a5 a7

Players I and II alternate playing numbers an ∈ ω,

1

Page 18: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8

II a1 a3 a5 a7 a9

Players I and II alternate playing numbers an ∈ ω,

1

Page 19: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8 · · ·II a1 a3 a5 a7 a9 · · ·

Players I and II alternate playing numbers an ∈ ω,forming tog-

ether an infinite sequence z = 〈a0, a1, a2, · · · · · · 〉 ∈ ωω.

If z belongs to A then player I wins.

If z does not belong to A then player II wins.

1

Page 20: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8 · · ·II a1 a3 a5 a7 a9 · · ·

Players I and II alternate playing numbers an ∈ ω, forming tog-

ether an infinite sequence z = 〈a0, a1, a2, · · · · · · 〉 ∈ ωω.

1

Page 21: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8 · · ·II a1 a3 a5 a7 a9 · · ·

Players I and II alternate playing numbers an ∈ ω, forming tog-

ether an infinite sequence z = 〈a0, a1, a2, · · · · · · 〉 ∈ ωω.

If z belongs to A then player I wins.

1

Page 22: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8 · · ·II a1 a3 a5 a7 a9 · · ·

Players I and II alternate playing numbers an ∈ ω, forming tog-

ether an infinite sequence z = 〈a0, a1, a2, · · · · · · 〉 ∈ ωω.

If z belongs to A then player I wins.

If z does not belong to A then player II wins.

1

Page 23: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Determinacy:

ωω is the set of infinite sequences of natural numbers.

Let A ⊂ ωω. Define Gω(A) to be the following game:

I a0 a2 a4 a6 a8 · · ·II a1 a3 a5 a7 a9 · · ·

Players I and II alternate playing numbers an ∈ ω, forming tog-

ether an infinite sequence z = 〈a0, a1, a2, · · · · · · 〉 ∈ ωω.

If z belongs to A then player I wins.

If z does not belong to A then player II wins.

Gω(A) is determined if one of the players has a winning strategy.

(A strategy is a complete recipe that instructs the player precisely

how to play in each conceivable situation.)

1

Page 24: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

2

Page 25: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

Using the axiom of choice (just a wellordering of R) it is easy to

construct a non-determined set.

2

Page 26: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

Using the axiom of choice (just a wellordering of R) it is easy to

construct a non-determined set.

det(P(ωω)) is therefore false.

2

Page 27: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

Using the axiom of choice (just a wellordering of R) it is easy to

construct a non-determined set.

det(P(ωω)) is therefore false.

But determinacy for definable sets is:

2

Page 28: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

Using the axiom of choice (just a wellordering of R) it is easy to

construct a non-determined set.

det(P(ωω)) is therefore false.

But determinacy for definable sets is: (1) true;

2

Page 29: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

Using the axiom of choice (just a wellordering of R) it is easy to

construct a non-determined set.

det(P(ωω)) is therefore false.

But determinacy for definable sets is: (1) true; and

2

Page 30: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are

determined.

Using the axiom of choice (just a wellordering of R) it is easy to

construct a non-determined set.

det(P(ωω)) is therefore false.

But determinacy for definable sets is: (1) true; and (2) useful.

2

Page 31: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

3

Page 32: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.

3

Page 33: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.

Ns (s ∈ ω<ω) are the basic open sets.

3

Page 34: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.

Ns (s ∈ ω<ω) are the basic open sets.

A ⊂ ωω is open if it is a union of basic open sets.

3

Page 35: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.

Ns (s ∈ ω<ω) are the basic open sets.

A ⊂ ωω is open if it is a union of basic open sets.

ωω with this topology is Baire space.

3

Page 36: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.

Ns (s ∈ ω<ω) are the basic open sets.

A ⊂ ωω is open if it is a union of basic open sets.

ωω with this topology is Baire space.

Following standard abuse of notation identify it with R.

3

Page 37: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.

Ns (s ∈ ω<ω) are the basic open sets.

A ⊂ ωω is open if it is a union of basic open sets.

3

Page 38: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

Ns = {x ∈ R | x extends s} (s ∈ ω<ω) are the basic open sets.

A ⊂ R is open if it is a union of basic open sets.

3

Page 39: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

Ns = {x ∈ R | x extends s} (s ∈ ω<ω) are the basic open sets.

A ⊂ R is open if it is a union of basic open sets.

The Borel sets are those that can be obtained from open sets

using complementations and countable unions.

3

Page 40: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

Ns = {x ∈ R | x extends s} (s ∈ ω<ω) are the basic open sets.

A ⊂ R is open if it is a union of basic open sets.

The Borel sets are those that can be obtained from open sets

using complementations and countable unions.

The projection of B ⊂ R × R is the set {x | (∃y)〈x, y〉 ∈ B}.

3

Page 41: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

Ns = {x ∈ R | x extends s} (s ∈ ω<ω) are the basic open sets.

A ⊂ R is open if it is a union of basic open sets.

The Borel sets are those that can be obtained from open sets

using complementations and countable unions.

The projection of B ⊂ R × R is the set {x | (∃y)〈x, y〉 ∈ B}.

A set is analytic if it is the projection of a closed set.

3

Page 42: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

Ns = {x ∈ R | x extends s} (s ∈ ω<ω) are the basic open sets.

A ⊂ R is open if it is a union of basic open sets.

The Borel sets are those that can be obtained from open sets

using complementations and countable unions.

The projection of B ⊂ R × R is the set {x | (∃y)〈x, y〉 ∈ B}.

A set is analytic if it is the projection of a closed set.

A set is projective if it can be obtained from an open set using

complementations and projections.

3

Page 43: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

ω<ω is the set of finite sequences of natural numbers.

Ns = {x ∈ R | x extends s} (s ∈ ω<ω) are the basic open sets.

A ⊂ R is open if it is a union of basic open sets.

The Borel sets are those that can be obtained from open sets

using complementations and countable unions.

The projection of B ⊂ R × R is the set {x | (∃y)〈x, y〉 ∈ B}.

A set is analytic if it is the projection of a closed set.

A set is projective if it can be obtained from an open set using

complementations and projections.

{Borel sets} ( {analytic sets} ( {projective sets}.

3

Page 44: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

4

Page 45: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

4

Page 46: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

4

Page 47: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

4

Page 48: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

L(R) is the smallest model of set theory which contains all the re-

als and all the ordinals.

4

Page 49: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

L(R) is the smallest model of set theory which contains all the re-

als and all the ordinals. It is obtained as the union⋃

α∈ON Lα(R)

where:

4

Page 50: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

L(R) is the smallest model of set theory which contains all the re-

als and all the ordinals. It is obtained as the union⋃

α∈ON Lα(R)

where: L0(R) = R,

4

Page 51: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

L(R) is the smallest model of set theory which contains all the re-

als and all the ordinals. It is obtained as the union⋃

α∈ON Lα(R)

where: L0(R) = R, Lλ(R) =⋃

α<λ Lα(R) for limit λ,

4

Page 52: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

L(R) is the smallest model of set theory which contains all the re-

als and all the ordinals. It is obtained as the union⋃

α∈ON Lα(R)

where: L0(R) = R, Lλ(R) =⋃

α<λ Lα(R) for limit λ, Lα+1(R) =

Lα(R) ∪ {A ⊂ Lα(R) | A is 1st order definable over Lα(R)}.

4

Page 53: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

L(R) is the smallest model of set theory which contains all the re-

als and all the ordinals. It is obtained as the union⋃

α∈ON Lα(R)

where: L0(R) = R, Lλ(R) =⋃

α<λ Lα(R) for limit λ, Lα+1(R) =

Lα(R) ∪ {A ⊂ Lα(R) | A is 1st order definable over Lα(R)}.

{projective sets} ⊂ L1(R).

4

Page 54: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

4

Page 55: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

Theorem 5 (Woodin 1985) All sets of reals in L(R) are de-

termined.

4

Page 56: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

Theorem 5 (Woodin 1985) All sets of reals in L(R) are de-

termined.

Theorems 1 and 2 are in ZFC, the basic system of axioms for

set theory.

4

Page 57: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 1 (Gale–Stewart 1953) All open sets are determined.

Theorem 2 (Martin 1975) All Borel sets are determined.

Theorem 3 (Martin 1970) All analytic sets are determined.

Theorem 4 (Martin–Steel 1985) All projective sets are deter-

mined.

Theorem 5 (Woodin 1985) All sets of reals in L(R) are de-

termined.

Theorems 1 and 2 are in ZFC, the basic system of axioms for

set theory.

Theorems 3, 4, and 5 require large cardinal axioms.

4

Page 58: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

5

Page 59: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

5

Page 60: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

5

Page 61: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ:

5

Page 62: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ.

5

Page 63: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ. ∃Γ is the pointclass of projections of sets in Γ.

5

Page 64: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ. ∃Γ is the pointclass of projections of sets in Γ.

Σ11 = {analytic sets}. Π

1n = ¬Σ

1n. Σ

1n+1 = ∃Π1

n.

5

Page 65: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ. ∃Γ is the pointclass of projections of sets in Γ.

Σ11 = {analytic sets}. Π

1n = ¬Σ

1n. Σ

1n+1 = ∃Π1

n. ∆1n = Π

1n ∩ Σ

1n.

5

Page 66: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ. ∃Γ is the pointclass of projections of sets in Γ.

Σ11 = {analytic sets}. Π

1n = ¬Σ

1n. Σ

1n+1 = ∃Π1

n. ∆1n = Π

1n ∩ Σ

1n.

Every Σ1n set A is obtained from an underlying open set using

negations and existential quantifiers.

5

Page 67: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ. ∃Γ is the pointclass of projections of sets in Γ.

Σ11 = {analytic sets}. Π

1n = ¬Σ

1n. Σ

1n+1 = ∃Π1

n. ∆1n = Π

1n ∩ Σ

1n.

Every Σ1n set A is obtained from an underlying open set using

negations and existential quantifiers. A is (lightface) Σ1n if the

underlying open set is recursive.

5

Page 68: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then

all sets in Γ have the Baire property.

Theorem 7 (Mycielski–Swierczkowski 1964) Assume det(Γ).

Then all sets in Γ are Lebesgue measurable.

Theorem 8 (Davis 1964) Assume det(Γ). Let A ∈ Γ. Then

either A is countable or else it contains a perfect set.

For a pointclass Γ: ¬Γ is the pointclass of complements of sets

in Γ. ∃Γ is the pointclass of projections of sets in Γ.

Σ11 = {analytic sets}. Π

1n = ¬Σ

1n. Σ

1n+1 = ∃Π1

n. ∆1n = Π

1n ∩ Σ

1n.

Every Σ1n set A is obtained from an underlying open set using

negations and existential quantifiers. A is (lightface) Σ1n if the

underlying open set is recursive. Similarly with Π1n.

5

Page 69: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Γ has the reduction property if for any A, B ∈ Γ, there are A′ ⊂ A,

B′ ⊂ B in Γ, so that A′ ∪ B′ = A ∪ B and A′ ∩ B′ = ∅.

6

Page 70: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Γ has the reduction property if for any A, B ∈ Γ, there are A′ ⊂ A,

B′ ⊂ B in Γ, so that A′ ∪ B′ = A ∪ B and A′ ∩ B′ = ∅.

Theorem 9 (Kuratowski 1936) The pointclasses Π11 and Σ

12

have the reduction property.

6

Page 71: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Γ has the reduction property if for any A, B ∈ Γ, there are A′ ⊂ A,

B′ ⊂ B in Γ, so that A′ ∪ B′ = A ∪ B and A′ ∩ B′ = ∅.

Theorem 9 (Kuratowski 1936) The pointclasses Π11 and Σ

12

have the reduction property.

Reduction for higher pointclasses cannot be settled in ZFC.

6

Page 72: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Γ has the reduction property if for any A, B ∈ Γ, there are A′ ⊂ A,

B′ ⊂ B in Γ, so that A′ ∪ B′ = A ∪ B and A′ ∩ B′ = ∅.

Theorem 9 (Kuratowski 1936) The pointclasses Π11 and Σ

12

have the reduction property.

Reduction for higher pointclasses cannot be settled in ZFC.

Blackwell (1967) obtained Π11 and Σ

12 with a very elegant argu-

ment using det(open).

6

Page 73: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Γ has the reduction property if for any A, B ∈ Γ, there are A′ ⊂ A,

B′ ⊂ B in Γ, so that A′ ∪ B′ = A ∪ B and A′ ∩ B′ = ∅.

Theorem 9 (Kuratowski 1936) The pointclasses Π11 and Σ

12

have the reduction property.

Reduction for higher pointclasses cannot be settled in ZFC.

Blackwell (1967) obtained Π11 and Σ

12 with a very elegant argu-

ment using det(open).

Inspired by his proof, Martin and Addison–Moschovakis proved

the reduction property for Π13, assuming det(∆1

2).

6

Page 74: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Γ has the reduction property if for any A, B ∈ Γ, there are A′ ⊂ A,

B′ ⊂ B in Γ, so that A′ ∪ B′ = A ∪ B and A′ ∩ B′ = ∅.

Theorem 9 (Kuratowski 1936) The pointclasses Π11 and Σ

12

have the reduction property.

Reduction for higher pointclasses cannot be settled in ZFC.

Blackwell (1967) obtained Π11 and Σ

12 with a very elegant argu-

ment using det(open).

Inspired by his proof, Martin and Addison–Moschovakis proved

the reduction property for Π13, assuming det(∆1

2).

In fact they did more. They obtained a fundamental property,

the prewellordering property, which implies reduction.

6

Page 75: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A prewellorder on A ⊂ R is a relation � on A which is transitive,

reflexive, and wellfounded.

7

Page 76: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A prewellorder on A ⊂ R is a relation � on A which is transitive,

reflexive, and wellfounded.

A pwo � induces an equivalence relation: x ∼ y iff x � y ∧ y � x.

The pwo gives rise to a wellorder of the equivalence classes.

7

Page 77: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A prewellorder on A ⊂ R is a relation � on A which is transitive,

reflexive, and wellfounded.

A pwo � induces an equivalence relation: x ∼ y iff x � y ∧ y � x.

The pwo gives rise to a wellorder of the equivalence classes.

� belongs to Γ if there are P , N in Γ, ¬Γ respectively, so that

for every y ∈ A, {x | x � y} = {x | 〈x, y〉 ∈ P} = {x | 〈x, y〉 ∈ N}.

7

Page 78: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A prewellorder on A ⊂ R is a relation � on A which is transitive,

reflexive, and wellfounded.

A pwo � induces an equivalence relation: x ∼ y iff x � y ∧ y � x.

The pwo gives rise to a wellorder of the equivalence classes.

� belongs to Γ if there are P , N in Γ, ¬Γ respectively, so that

for every y ∈ A, {x | x � y} = {x | 〈x, y〉 ∈ P} = {x | 〈x, y〉 ∈ N}.

Γ has the prewellordering property if every A ∈ Γ admits a

prewellorder in Γ.

7

Page 79: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A prewellorder on A ⊂ R is a relation � on A which is transitive,

reflexive, and wellfounded.

A pwo � induces an equivalence relation: x ∼ y iff x � y ∧ y � x.

The pwo gives rise to a wellorder of the equivalence classes.

� belongs to Γ if there are P , N in Γ, ¬Γ respectively, so that

for every y ∈ A, {x | x � y} = {x | 〈x, y〉 ∈ P} = {x | 〈x, y〉 ∈ N}.

Γ has the prewellordering property if every A ∈ Γ admits a

prewellorder in Γ.

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

7

Page 80: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

8

Page 81: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

8

Page 82: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

8

Page 83: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

8

Page 84: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

8

Page 85: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

Think of a as a quantifier, somewhere between ∀ and ∃.

8

Page 86: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

Think of a as a quantifier, somewhere between ∀ and ∃.

(∃y)〈x, y〉 ∈ B ⇐⇒ (∃y(0))(∃y(1))(∃y(2)) · · · · · · 〈x, y〉 ∈ B.

8

Page 87: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

Think of a as a quantifier, somewhere between ∀ and ∃.

8

Page 88: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

Think of a as a quantifier, somewhere between ∀ and ∃.

(∀y)〈x, y〉 ∈ B ⇐⇒ (∀y(0))(∀y(1))(∀y(2)) · · · · · · 〈x, y〉 ∈ B.

8

Page 89: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

Think of a as a quantifier, somewhere between ∀ and ∃.

8

Page 90: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Write (ay)〈x, y〉 ∈ B for x ∈ aB.

Think of a as a quantifier, somewhere between ∀ and ∃.

(ay)〈x, y〉 ∈ B ⇐⇒ (∃y(0))(∀y(1))(∃y(2)) · · · · · · 〈x, y〉 ∈ B.

8

Page 91: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

8

Page 92: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Easy to check aΠ1n = Σ

1n+1, and (using determinacy) aΣ

1n =

Π1n+1.

8

Page 93: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 (Martin, Addison–Moschovakis 1968) Assume

det(projective). Then the projective pointclasses with the pwo

property, and similarly reduction, are Π11, Σ

12, Π

13, Σ

14, · · · · · · .

For B ⊂ R × R set Bx = {y | 〈x, y〉 ∈ B}.

Set aB = {x | I has a w.s. in Gω(Bx)}.

For a pointclass Γ set aΓ = {aB | B ∈ Γ}.

Easy to check aΠ1n = Σ

1n+1, and (using determinacy) aΣ

1n =

Π1n+1.

The pointclasses in Theorem 10 are therefore precisely the point-

classes a(n)Π11, n < ω.

8

Page 94: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

9

Page 95: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

The axiom of determinacy (AD), stating that all sets of reals are

determined, became standard in the study of L(R).

9

Page 96: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

The axiom of determinacy (AD), stating that all sets of reals are

determined, became standard in the study of L(R).

Let δ denote the supremum of the lengths of ∆ pwos on ∆ sets.

9

Page 97: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

The axiom of determinacy (AD), stating that all sets of reals are

determined, became standard in the study of L(R).

Let δ denote the supremum of the lengths of ∆ pwos on ∆ sets.

Theorem 11 Assume AD. Then δ11 = ω1, δ12 = ω2 (Martin), and

δ13 = ωω+1 (Martin). (Much more known.)

9

Page 98: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

The axiom of determinacy (AD), stating that all sets of reals are

determined, became standard in the study of L(R).

Let δ denote the supremum of the lengths of ∆ pwos on ∆ sets.

Theorem 11 Assume AD. Then δ11 = ω1, δ12 = ω2 (Martin), and

δ13 = ωω+1 (Martin). (Much more known.)

Values of δ1n are absolute between L(R) and V.

9

Page 99: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

The axiom of determinacy (AD), stating that all sets of reals are

determined, became standard in the study of L(R).

Let δ denote the supremum of the lengths of ∆ pwos on ∆ sets.

Theorem 11 Assume AD. Then δ11 = ω1, δ12 = ω2 (Martin), and

δ13 = ωω+1 (Martin). (Much more known.)

Values of δ1n are absolute between L(R) and V. So, e.g., δ12 =

(ω2)L(R) assuming AD

L(R).

9

Page 100: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 10 helped establish determinacy as the right assump-

tion in the study of definable sets.

The axiom of determinacy (AD), stating that all sets of reals are

determined, became standard in the study of L(R).

Let δ denote the supremum of the lengths of ∆ pwos on ∆ sets.

Theorem 11 Assume AD. Then δ11 = ω1, δ12 = ω2 (Martin), and

δ13 = ωω+1 (Martin). (Much more known.)

Values of δ1n are absolute between L(R) and V. So, e.g., δ12 =

(ω2)L(R) assuming AD

L(R).

Theorem 12 (Steel–Van Wesep–Woodin) Assume ADL(R).

Then it is consistent (with ADL(R) and AC) that (ω2)

L(R) = ω2,

and hence δ12 = ω2.

9

Page 101: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

10

Page 102: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

10

Page 103: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

10

Page 104: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

κ must be a cardinal.

10

Page 105: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

κ must be a cardinal. Otherwise have τ < κ and a surjection

f : τ → κ. But then by elementarity π(f) is onto π(κ). Since

f ⊂ τ × κ ⊂ crit(π)2, π(f) = f . So π(κ) = κ, contradiction.

10

Page 106: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

κ must be a cardinal. Otherwise have τ < κ and a surjection

f : τ → κ. But then by elementarity π(f) is onto π(κ). Since

f ⊂ τ × κ ⊂ crit(π)2, π(f) = f . So π(κ) = κ, contradiction.

κ must be a limit cardinal.

10

Page 107: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

κ must be a cardinal. Otherwise have τ < κ and a surjection

f : τ → κ. But then by elementarity π(f) is onto π(κ). Since

f ⊂ τ × κ ⊂ crit(π)2, π(f) = f . So π(κ) = κ, contradiction.

κ must be a limit cardinal. Otherwise have τ < κ so that κ = τ+.

But then by elementarity π(κ) = (π(τ)+)M . Yet π(τ) = τ , so

π(κ) = (τ+)M = κ, contradiction.

10

Page 108: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

κ must be a cardinal. Otherwise have τ < κ and a surjection

f : τ → κ. But then by elementarity π(f) is onto π(κ). Since

f ⊂ τ × κ ⊂ crit(π)2, π(f) = f . So π(κ) = κ, contradiction.

κ must be a limit cardinal. Otherwise have τ < κ so that κ = τ+.

But then by elementarity π(κ) = (π(τ)+)M . Yet π(τ) = τ , so

π(κ) = (τ+)M = κ, contradiction.

Similar arguments show κ must be inaccessible, and in fact can-

not be described from below in any absolute manner.

10

Page 109: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Large cardinals:

Large cardinal axioms state the existence of (non-trivial) elemen-

tary embeddings π : V → M ⊂ V.

The critical point of π is the first ordinal κ so that π(κ) 6= κ.

κ must be a cardinal. Otherwise have τ < κ and a surjection

f : τ → κ. But then by elementarity π(f) is onto π(κ). Since

f ⊂ τ × κ ⊂ crit(π)2, π(f) = f . So π(κ) = κ, contradiction.

κ must be a limit cardinal. Otherwise have τ < κ so that κ = τ+.

But then by elementarity π(κ) = (π(τ)+)M . Yet π(τ) = τ , so

π(κ) = (τ+)M = κ, contradiction.

Similar arguments show κ must be inaccessible, and in fact can-

not be described from below in any absolute manner.

So the existence of non-trivial π : V → M ⊂ V cannot be proved

in ZFC, and the first ordinal moved by π must be very large.

10

Page 110: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

11

Page 111: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

(Using an ultrapower construction, the measurability of κ is

equivalent to the existence of a total, non-principal, countably

complete, 2-valued measure on κ.)

11

Page 112: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

11

Page 113: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

11

Page 114: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

11

Page 115: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

11

Page 116: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ,

11

Page 117: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ, and λ–strong wrt

D if in addition λ ∩ π(D) = λ ∩ D.

11

Page 118: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ, and λ–strong wrt

D if in addition λ ∩ π(D) = λ ∩ D.

κ is <δ–strong if it is the critical point of a λ–strong embedding

for each λ < δ.

11

Page 119: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ, and λ–strong wrt

D if in addition λ ∩ π(D) = λ ∩ D.

κ is <δ–strong if it is the critical point of a λ–strong embedding

for each λ < δ. Similarly wrt D.

11

Page 120: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ, and λ–strong wrt

D if in addition λ ∩ π(D) = λ ∩ D.

κ is <δ–strong if it is the critical point of a λ–strong embedding

for each λ < δ. Similarly wrt D.

Note: if κ is the first measurable cardinal, then κ is only κ+–

strong.

11

Page 121: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ, and λ–strong wrt

D if in addition λ ∩ π(D) = λ ∩ D.

κ is <δ–strong if it is the critical point of a λ–strong embedding

for each λ < δ. Similarly wrt D.

11

Page 122: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A cardinal is measurable if it is the critical point of an embedding

π : V → M ⊂ V.

Theorem (Martin) Suppose there is a measurable cardinals.

Then all Π11 sets are determined.

π : L → L is enough (M.), and also necessary (Harrington).

Greater strength from π : V → M can be obtained by demanding

agreement between M and V.

π is λ–strong if M has all bounded subsets of λ, and λ–strong wrt

D if in addition λ ∩ π(D) = λ ∩ D.

κ is <δ–strong if it is the critical point of a λ–strong embedding

for each λ < δ. Similarly wrt D.

δ is a Woodin cardinal if for every D ⊂ δ there is κ < δ which is

<δ–strong wrt D.

11

Page 123: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

12

Page 124: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

From E one can construct (using ultrapowers) an embedding

σ : V → Ult(V, E) which agrees with π to λ, meaning that σ(X)∩

λ = π(X) ∩ λ.

12

Page 125: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

From E one can construct (using ultrapowers) an embedding

σ : V → Ult(V, E) which agrees with π to λ, meaning that σ(X)∩

λ = π(X) ∩ λ. (σ is called the ultrapower embedding.)

12

Page 126: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

From E one can construct (using ultrapowers) an embedding

σ : V → Ult(V, E) which agrees with π to λ, meaning that σ(X)∩

λ = π(X) ∩ λ. (σ is called the ultrapower embedding.)

Extenders are thus enough to capture strength as defined above.

12

Page 127: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

From E one can construct (using ultrapowers) an embedding

σ : V → Ult(V, E) which agrees with π to λ, meaning that σ(X)∩

λ = π(X) ∩ λ. (σ is called the ultrapower embedding.)

Extenders are thus enough to capture strength as defined above.

Models Q, Q∗ agree past κ if PQ∗(κ) = PQ(κ).

12

Page 128: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

From E one can construct (using ultrapowers) an embedding

σ : V → Ult(V, E) which agrees with π to λ, meaning that σ(X)∩

λ = π(X) ∩ λ. (σ is called the ultrapower embedding.)

Extenders are thus enough to capture strength as defined above.

Models Q, Q∗ agree past κ if PQ∗(κ) = PQ(κ).

If E is an extender with critical point κ in a model Q, and Q∗

agrees with Q past κ, then E gives rise, again using ultrapowers,

to an embedding acting on Q∗.

12

Page 129: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let π : V → M . Let κ = crit(π) and λ ≤ π(κ). The (κ, λ)–

extender induced by π is the function E : P(κ) → P(λ) defined

by E(X) = π(X) ∩ λ.

From E one can construct (using ultrapowers) an embedding

σ : V → Ult(V, E) which agrees with π to λ, meaning that σ(X)∩

λ = π(X) ∩ λ. (σ is called the ultrapower embedding.)

Extenders are thus enough to capture strength as defined above.

Models Q, Q∗ agree past κ if PQ∗(κ) = PQ(κ).

If E is an extender with critical point κ in a model Q, and Q∗

agrees with Q past κ, then E gives rise, again using ultrapowers,

to an embedding acting on Q∗.

This allows constructing iterated ultrapowers with non-linear

base orders.

12

Page 130: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 131: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 132: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 133: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 134: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 135: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 136: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 137: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 138: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 139: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 140: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

The creation of an iteration tree on M requires large cardinals

in M .

14

Page 141: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

The creation of an iteration tree on M requires large cardinals

in M .

For non-linear iterations, measurable cardinals are not enough.

14

Page 142: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

The creation of an iteration tree on M requires large cardinals

in M .

For non-linear iterations, measurable cardinals are not enough.

If Ek ∈ Mk is to be applied to Ml for some l < k, then Ml and

Mk must agree past crit(Ek).

14

Page 143: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

The creation of an iteration tree on M requires large cardinals

in M .

For non-linear iterations, measurable cardinals are not enough.

If Ek ∈ Mk is to be applied to Ml for some l < k, then Ml and

Mk must agree past crit(Ek).

Thus the extenders used to form the models between Ml and Mk

must be crit(Ek)–strong.

14

Page 144: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

The creation of an iteration tree on M requires large cardinals

in M .

For non-linear iterations, measurable cardinals are not enough.

If Ek ∈ Mk is to be applied to Ml for some l < k, then Ml and

Mk must agree past crit(Ek).

Thus the extenders used to form the models between Ml and Mk

must be crit(Ek)–strong.

The creation of iteration trees with several cofinal branches re-

quires many strong extenders.

14

Page 145: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

The creation of an iteration tree on M requires large cardinals

in M .

For non-linear iterations, measurable cardinals are not enough.

If Ek ∈ Mk is to be applied to Ml for some l < k, then Ml and

Mk must agree past crit(Ek).

Thus the extenders used to form the models between Ml and Mk

must be crit(Ek)–strong.

The creation of iteration trees with several cofinal branches re-

quires many strong extenders.

Woodin cardinals give precisely the extenders needed.

14

Page 146: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

In fact using the extenders given by Woodin cardinals one can

construct iteration trees very flexibly, reducing quantifiers over

real numbers to quantifiers over iteration trees and branches

through them.

15

Page 147: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

In fact using the extenders given by Woodin cardinals one can

construct iteration trees very flexibly, reducing quantifiers over

real numbers to quantifiers over iteration trees and branches

through them.

Such reductions are key to:

15

Page 148: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

In fact using the extenders given by Woodin cardinals one can

construct iteration trees very flexibly, reducing quantifiers over

real numbers to quantifiers over iteration trees and branches

through them.

Such reductions are key to:

Theorem (Martin–Steel) Suppose there are n Woodin cardi-

nals and a measurable cardinal above them. Then all Π1n+1 sets

are determined.

15

Page 149: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

In fact using the extenders given by Woodin cardinals one can

construct iteration trees very flexibly, reducing quantifiers over

real numbers to quantifiers over iteration trees and branches

through them.

Such reductions are key to:

Theorem (Martin–Steel) Suppose there are n Woodin cardi-

nals and a measurable cardinal above them. Then all Π1n+1 sets

are determined.

Theorem (Woodin) Suppose there are ω Woodin cardinals and

a measurable cardinal above them. Then all sets in L(R) are

determined.

15

Page 150: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

In fact using the extenders given by Woodin cardinals one can

construct iteration trees very flexibly, reducing quantifiers over

real numbers to quantifiers over iteration trees and branches

through them.

Such reductions are key to:

Theorem (Martin–Steel) Suppose there are n Woodin cardi-

nals and a measurable cardinal above them. Then all Π1n+1 sets

are determined.

Theorem (Woodin) Suppose there are ω Woodin cardinals and

a measurable cardinal above them. Then all sets in L(R) are

determined.

In both cases Woodin cardinals in iterable inner models (rather

than the actual universe V) are enough, and moreover necessary.

15

Page 151: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

These theorems are the starting point for a very strong connec-

tion between large cardinals and the theory of L(R).

16

Page 152: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

These theorems are the starting point for a very strong connec-

tion between large cardinals and the theory of L(R).

Many of the results on L(R) can be proved from determinacy

(which in turn is proved from large cardinals).

16

Page 153: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

These theorems are the starting point for a very strong connec-

tion between large cardinals and the theory of L(R).

Many of the results on L(R) can be proved from determinacy

(which in turn is proved from large cardinals).

But some make direct use of inner models for Woodin cardinals.

16

Page 154: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

17

Page 155: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

(The hierarchy generated by this definition is the difference hi-

erarchy on Π11 sets. If α = 2 for example, then the condition

states simply that A = A0 − A1.)

17

Page 156: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

17

Page 157: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

17

Page 158: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

17

Page 159: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

For n = 0: Π11 determinacy gives a non-trivial π : L → L (Har-

rington), which in turn gives <ω2–Π11 determinacy (Martin).

17

Page 160: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

For n = 0: Π11 determinacy gives a non-trivial π : L → L (Har-

rington), which in turn gives <ω2–Π11 determinacy (Martin).

Generally: Π1n+1 determinacy gives non-trivial π : M → M where

M is an iterable class model with n Woodin cardinals (Woodin),

which in turn gives a(n)(<ω2–Π11) determinacy (Neeman).

17

Page 161: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

For n = 0: Π11 determinacy gives a non-trivial π : L → L (Har-

rington), which in turn gives <ω2–Π11 determinacy (Martin).

Generally: Π1n+1 determinacy gives non-trivial π : M → M where

M is an iterable class model with n Woodin cardinals (Woodin),

which in turn gives a(n)(<ω2–Π11) determinacy (Neeman).

Theorem known previously for odd n, not using large cardinals

(Kechris–Woodin).

17

Page 162: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

17

Page 163: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

Theorem 14 (Hjorth) Work in L(R) assuming AD. Let � be a

a(α–Π11) prewellorder with α < ω · k. Then the ordertype of � is

smaller than ωk+1.

17

Page 164: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

A set A is α–Π11 if there is a sequence 〈Aξ | ξ < α〉 of Π

11 sets so

that x ∈ A iff the least ξ so that x 6∈ Aξ ∨ ξ = α is odd.

The lightface notion is defined similarly, requiring a recursive

code for the sequence.

Theorem 13 (Neeman–Woodin) det(Π1n+1) implies determi-

nacy for all sets in the larger pointclass a(n)(<ω2–Π11).

Theorem 14 (Hjorth) Work in L(R) assuming AD. Let � be a

a(α–Π11) prewellorder with α < ω · k. Then the ordertype of � is

smaller than ωk+1.

Theorem 15 (Neeman, Woodin) Assume ADL(R). Then it is

consistent (with ADL(R) and the axiom of choice) that δ

13 = ω2.

17

Page 165: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

18

Page 166: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Iterable:

18

Page 167: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Iterable:

The creation of iteration trees requires some choice at limits.

18

Page 168: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Constructed in stages, starting

from a base model M = M0.

E.g., having constructed

M1, . . . , M6: pick an extender

E6 ∈ M6, apply it to M1,

setting M7 = Ult(M1, E6) and

letting j1,7 : M1 → M7 be the

ultrapower embedding.

At limit λ: pick a branch

through the tree, cofinal in λ.

Set Mλ equal to the direct limit

of models and embeddings along

this branch.

The result is an iteration tree on

M .

Mω+1

M7

M6

M5

M4

M3

M2

j2,4

M1

j1,3

M0

j0,2

j0,1

13

Page 169: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Iterable:

The creation of iteration trees requires some choice at limits.

18

Page 170: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Iterable:

The creation of iteration trees requires some choice at limits.

M is iterable if these choices can be made in a way that secures

the wellfoundedness of all the models created.

18

Page 171: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

18

Page 172: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Minimal:

18

Page 173: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Minimal:

A model consisting of just the sets constructible from enough

extenders to witness θ.

18

Page 174: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Minimal:

A model consisting of just the sets constructible from enough

extenders to witness θ.

Minimal for θ in much the same way L is minimal for ZFC.

18

Page 175: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Minimal:

A model consisting of just the sets constructible from enough

extenders to witness θ.

Minimal for θ in much the same way L is minimal for ZFC.

Iterability crucial for making sense of minimality in the presence

of extenders.

18

Page 176: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Minimal:

A model consisting of just the sets constructible from enough

extenders to witness θ.

Minimal for θ in much the same way L is minimal for ZFC.

Iterability crucial for making sense of minimality in the presence

of extenders.

Comparisons through iterated ultrapowers show that any two

ways to witness θ are compatible.

18

Page 177: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

18

Page 178: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Let κ = crit(π). The theory of the sharp for θ is⊕

k<ω Tk where

Tk is the theory of 〈κ, π(κ), · · · , πk−1(κ)〉 in M .

18

Page 179: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Let κ = crit(π). The theory of the sharp for θ is⊕

k<ω Tk where

Tk is the theory of 〈κ, π(κ), · · · , πk−1(κ)〉 in M .

The sharp for “v = v” is called 0♯. It is a non-trivial π : L → L.

18

Page 180: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Let κ = crit(π). The theory of the sharp for θ is⊕

k<ω Tk where

Tk is the theory of 〈κ, π(κ), · · · , πk−1(κ)〉 in M .

The sharp for “v = v” is called 0♯. It is a non-trivial π : L → L.

Theorem 16 (Martin) Let Bi be a recursive enumeration of

the <ω2–Π11 sets. Suppose 0♯ exists. Then all <ω2–Π1

1 games

are determined, and {i | I has a w.s. in Gω(Bi)} is recursively

isomorphic to the theory of 0♯.

18

Page 181: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let θ(v) be a formula. A sharp for θ is a non-trivial embedding

π : M → M where M is the minimal iterable class model admitting

a non-trivial embedding π and satisfying θ[crit(π)].

Let κ = crit(π). The theory of the sharp for θ is⊕

k<ω Tk where

Tk is the theory of 〈κ, π(κ), · · · , πk−1(κ)〉 in M .

The sharp for “v = v” is called 0♯. It is a non-trivial π : L → L.

Theorem 16 (Martin) Let Bi be a recursive enumeration of

the <ω2–Π11 sets. Suppose 0♯ exists. Then all <ω2–Π1

1 games

are determined, and {i | I has a w.s. in Gω(Bi)} is recursively

isomorphic to the theory of 0♯.

Theorem 17 (Neeman) Let Bi be a recursive enumeration of

the a(n)(<ω2–Π11) sets. Suppose a sharp for n Woodin cardinals

exists. Then all a(n)(<ω2–Π11) games are determined, and {i | I

has a w.s. in Gω(Bi)} is recursively isomorphic to the theory of

the sharp for n Woodin cardinals.

18

Page 182: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 16 (Martin) Let Bi be a recursive enumeration of

the <ω2–Π11 sets. Suppose 0♯ exists. Then all <ω2–Π1

1 games

are determined, and {i | I has a w.s. in Gω(Bi)} is recursively

isomorphic to the theory of 0♯.

Theorem 17 (Neeman) Let Bi be a recursive enumeration of

the a(n)(<ω2–Π11) sets. Suppose a sharp for n Woodin cardinals

exists. Then all a(n)(<ω2–Π11) games are determined, and {i | I

has a w.s. in Gω(Bi)} is recursively isomorphic to the theory of

the sharp for n Woodin cardinals.

19

Page 183: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 16 (Martin) Let Bi be a recursive enumeration of

the <ω2–Π11 sets. Suppose 0♯ exists. Then all <ω2–Π1

1 games

are determined, and {i | I has a w.s. in Gω(Bi)} is recursively

isomorphic to the theory of 0♯.

Theorem 17 (Neeman) Let Bi be a recursive enumeration of

the a(n)(<ω2–Π11) sets. Suppose a sharp for n Woodin cardinals

exists. Then all a(n)(<ω2–Π11) games are determined, and {i | I

has a w.s. in Gω(Bi)} is recursively isomorphic to the theory of

the sharp for n Woodin cardinals.

These theorems give tight connection between the theory of em-

beddings acting on models for large cardinals, and determinacy.

19

Page 184: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 16 (Martin) Let Bi be a recursive enumeration of

the <ω2–Π11 sets. Suppose 0♯ exists. Then all <ω2–Π1

1 games

are determined, and {i | I has a w.s. in Gω(Bi)} is recursively

isomorphic to the theory of 0♯.

Theorem 17 (Neeman) Let Bi be a recursive enumeration of

the a(n)(<ω2–Π11) sets. Suppose a sharp for n Woodin cardinals

exists. Then all a(n)(<ω2–Π11) games are determined, and {i | I

has a w.s. in Gω(Bi)} is recursively isomorphic to the theory of

the sharp for n Woodin cardinals.

These theorems give tight connection between the theory of em-

beddings acting on models for large cardinals, and determinacy.

The connection (with analogues for ω Woodin cardinals) is cru-

cial for Theorems 13–15.

19

Page 185: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Analogous connections exist for stronger large cardinal axioms,

and stronger forms of determinacy.

20

Page 186: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Analogous connections exist for stronger large cardinal axioms,

and stronger forms of determinacy.

Reach as far as games of length ω1.

20

Page 187: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Analogous connections exist for stronger large cardinal axioms,

and stronger forms of determinacy.

Reach as far as games of length ω1.

Let ~S = 〈Sa | a ∈ [ω1]<ω〉 be a collection of mutually disjoint

stationary subsets of ω1.

20

Page 188: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Analogous connections exist for stronger large cardinal axioms,

and stronger forms of determinacy.

Reach as far as games of length ω1.

Let ~S = 〈Sa | a ∈ [ω1]<ω〉 be a collection of mutually disjoint

stationary subsets of ω1.

(With a stationary set Sa associated to each tuple a ∈ [ω1]<ω.)

20

Page 189: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Analogous connections exist for stronger large cardinal axioms,

and stronger forms of determinacy.

Reach as far as games of length ω1.

Let ~S = 〈Sa | a ∈ [ω1]<ω〉 be a collection of mutually disjoint

stationary subsets of ω1.

(With a stationary set Sa associated to each tuple a ∈ [ω1]<ω.)

Let [~S] denote the set

{〈α0, . . . , αk−1〉 ∈ [ω1]<ω | (∀i < k) αi ∈ S〈α0,...,αi−1〉

}.

20

Page 190: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let ϕ(x0, . . . , xk−1) be a formula.

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Page 191: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let ϕ(x0, . . . , xk−1) be a formula. Define Gω1(~S, ϕ) to be the

following game:

21

Page 192: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let ϕ(x0, . . . , xk−1) be a formula. Define Gω1(~S, ϕ) to be the

following game:

Players I and II alternate playing ω1 natural numbers, producing

together a sequence r ∈ ωω1.

21

Page 193: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let ϕ(x0, . . . , xk−1) be a formula. Define Gω1(~S, ϕ) to be the

following game:

Players I and II alternate playing ω1 natural numbers, producing

together a sequence r ∈ ωω1.

If there is a club C ⊂ ω1 so that (Lω1[r]; r) |= ϕ[α0, . . . , αk−1] for

all 〈α0, . . . , αk−1〉 ∈ [~S] ∩ [C]k then player I wins the run r.

21

Page 194: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let ϕ(x0, . . . , xk−1) be a formula. Define Gω1(~S, ϕ) to be the

following game:

Players I and II alternate playing ω1 natural numbers, producing

together a sequence r ∈ ωω1.

If there is a club C ⊂ ω1 so that (Lω1[r]; r) |= ϕ[α0, . . . , αk−1] for

all 〈α0, . . . , αk−1〉 ∈ [~S] ∩ [C]k then player I wins the run r.

If there is a club C ⊂ ω1 so that (Lω1[r]; r) |= ¬ϕ[α0, . . . , αk−1]

for all 〈α0, . . . , αk−1〉 ∈ [~S] ∩ [C]k then player II wins r.

21

Page 195: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Let ϕ(x0, . . . , xk−1) be a formula. Define Gω1(~S, ϕ) to be the

following game:

Players I and II alternate playing ω1 natural numbers, producing

together a sequence r ∈ ωω1.

If there is a club C ⊂ ω1 so that (Lω1[r]; r) |= ϕ[α0, . . . , αk−1] for

all 〈α0, . . . , αk−1〉 ∈ [~S] ∩ [C]k then player I wins the run r.

If there is a club C ⊂ ω1 so that (Lω1[r]; r) |= ¬ϕ[α0, . . . , αk−1]

for all 〈α0, . . . , αk−1〉 ∈ [~S] ∩ [C]k then player II wins r.

If neither condition holds then both players lose.

21

Page 196: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 18 (Neeman) Let ϕi be a recursive enumeration of

formulae. Suppose that these is a sharp π : M → M for the

statement “crit(π) is a Woodin cardinal.” Then:

22

Page 197: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 18 (Neeman) Let ϕi be a recursive enumeration of

formulae. Suppose that these is a sharp π : M → M for the

statement “crit(π) is a Woodin cardinal.” Then:

1. The games Gω1(~S, ϕ) are all determined.

22

Page 198: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 18 (Neeman) Let ϕi be a recursive enumeration of

formulae. Suppose that these is a sharp π : M → M for the

statement “crit(π) is a Woodin cardinal.” Then:

1. The games Gω1(~S, ϕ) are all determined.

2. Which player has a w.s. in Gω1(~S, ϕ) depends only on ϕ, not

on ~S.

22

Page 199: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 18 (Neeman) Let ϕi be a recursive enumeration of

formulae. Suppose that these is a sharp π : M → M for the

statement “crit(π) is a Woodin cardinal.” Then:

1. The games Gω1(~S, ϕ) are all determined.

2. Which player has a w.s. in Gω1(~S, ϕ) depends only on ϕ, not

on ~S.

3. The set {i | I has a w.s. in Gω1(~S, ϕi)} is recursively iso-

morphic to the theory of the sharp for “crit(π) is a Woodin

cardinal.”

22

Page 200: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Theorem 18 (Neeman) Let ϕi be a recursive enumeration of

formulae. Suppose that these is a sharp π : M → M for the

statement “crit(π) is a Woodin cardinal.” Then:

1. The games Gω1(~S, ϕ) are all determined.

2. Which player has a w.s. in Gω1(~S, ϕ) depends only on ϕ, not

on ~S.

3. The set {i | I has a w.s. in Gω1(~S, ϕi)} is recursively iso-

morphic to the theory of the sharp for “crit(π) is a Woodin

cardinal.”

The theorem establishes a precise analogue of Theorems 16 and

17, but for embeddings concentrating on Woodin cardinals and

for games of length ω1.

22

Page 201: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Question How high in the large cardinal hierarchy can such tight

connections between games and the theories of embeddings be

found?

23

Page 202: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Question How high in the large cardinal hierarchy can such tight

connections between games and the theories of embeddings be

found?

Theorem 18 is the frontier right now. But the large cardinals

involved are still low compared, for example, to superstrong.

23

Page 203: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Question How high in the large cardinal hierarchy can such tight

connections between games and the theories of embeddings be

found?

Theorem 18 is the frontier right now. But the large cardinals

involved are still low compared, for example, to superstrong.

Question What kind of games are tied to axioms higher up in

the large cardinal hierarchy?

23

Page 204: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Question How high in the large cardinal hierarchy can such tight

connections between games and the theories of embeddings be

found?

Theorem 18 is the frontier right now. But the large cardinals

involved are still low compared, for example, to superstrong.

Question What kind of games are tied to axioms higher up in

the large cardinal hierarchy?

Games motivated by Theorem 18 were used by Woodin in results

on Σ22 absoluteness.

23

Page 205: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Question How high in the large cardinal hierarchy can such tight

connections between games and the theories of embeddings be

found?

Theorem 18 is the frontier right now. But the large cardinals

involved are still low compared, for example, to superstrong.

Question What kind of games are tied to axioms higher up in

the large cardinal hierarchy?

Games motivated by Theorem 18 were used by Woodin in results

on Σ22 absoluteness. Other games similar to those in the theorem

are enough to capture the theory of superstrong cardinals.

23

Page 206: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

Question How high in the large cardinal hierarchy can such tight

connections between games and the theories of embeddings be

found?

Theorem 18 is the frontier right now. But the large cardinals

involved are still low compared, for example, to superstrong.

Question What kind of games are tied to axioms higher up in

the large cardinal hierarchy?

Games motivated by Theorem 18 were used by Woodin in results

on Σ22 absoluteness. Other games similar to those in the theorem

are enough to capture the theory of superstrong cardinals. But

there are no determinacy proofs for these games from large

cardinals, and indeed there are some negative results (Larson).

23

Page 207: Determinacy and large cardinalsineeman/ICM.pdfDeterminacy and Large Cardinals Itay Neeman Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555

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