View
3
Download
0
Category
Preview:
Citation preview
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
Determinacy:
1
Determinacy:
ωω is the set of infinite sequences of natural numbers.
1
Determinacy:
ωω is the set of infinite sequences of natural numbers.
Let A ⊂ ωω.
1
Determinacy:
ωω is the set of infinite sequences of natural numbers.
Let A ⊂ ωω. Define Gω(A) to be the following game:
1
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
For Γ ⊂ P(ωω), det(Γ) is the statement that all sets in Γ are
determined.
2
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
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
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
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
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
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
ω<ω is the set of finite sequences of natural numbers.
3
ω<ω is the set of finite sequences of natural numbers.
For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}.
3
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
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
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
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
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
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
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
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
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
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
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
Theorem 1 (Gale–Stewart 1953) All open sets are determined.
4
Theorem 1 (Gale–Stewart 1953) All open sets are determined.
Theorem 2 (Martin 1975) All Borel sets are determined.
4
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
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
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
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
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
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
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
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
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
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
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
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
Theorem 6 (Banach, Oxtoby 1957) Assume det(Γ). Then
all sets in Γ have the Baire property.
5
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
A prewellorder on A ⊂ R is a relation � on A which is transitive,
reflexive, and wellfounded.
7
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Theorem 10 helped establish determinacy as the right assump-
tion in the study of definable sets.
9
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
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
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
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
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
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
Large cardinals:
10
Large cardinals:
Large cardinal axioms state the existence of (non-trivial) elemen-
tary embeddings π : V → M ⊂ V.
10
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
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
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
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
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
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
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
A cardinal is measurable if it is the critical point of an embedding
π : V → M ⊂ V.
11
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
A cardinal is measurable if it is the critical point of an embedding
π : V → M ⊂ V.
11
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
The creation of an iteration tree on M requires large cardinals
in M .
14
The creation of an iteration tree on M requires large cardinals
in M .
For non-linear iterations, measurable cardinals are not enough.
14
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
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
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
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
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
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
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
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
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
These theorems are the starting point for a very strong connec-
tion between large cardinals and the theory of L(R).
16
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Mω
M7
M6
M5
M4
M3
M2
j2,4
M1
j1,3
M0
j0,2
j0,1
13
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Analogous connections exist for stronger large cardinal axioms,
and stronger forms of determinacy.
20
Analogous connections exist for stronger large cardinal axioms,
and stronger forms of determinacy.
Reach as far as games of length ω1.
20
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
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
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
Let ϕ(x0, . . . , xk−1) be a formula.
21
Let ϕ(x0, . . . , xk−1) be a formula. Define Gω1(~S, ϕ) to be the
following game:
21
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
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
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
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
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
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
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
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
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
Question How high in the large cardinal hierarchy can such tight
connections between games and the theories of embeddings be
found?
23
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
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
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
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
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
a
The End
Press Esc.
24
Recommended