A Journey Through the World of Mice and Games Projective and Beyond - - PowerPoint PPT Presentation

A Journey Through the World of Mice and Games

Projective and Beyond Sandra Uhlenbrock June 13th, 2016

Young Set Theory Workshop Copenhagen

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 1 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 2 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 3 / 29

Games in Set Theory

Definition (Gale/Stewart 1953)

Let A ⊂ 2N. With G(A) we denote the following game I i0 i2 . . . II i1 i3 . . . for in ∈ {0, 1} and n ∈ N. We say player I wins the game iff (in)n∈N ∈ A. Otherwise player II wins. We say G(A) (or A itself) is determined iff one of the players has a winning strategy (in the obvious sense).

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 4 / 29

Which games are determined?

Theorem (Gale/Stewart, 1953)

(AC) Let A ⊂ 2N be open or closed. Then G(A) is determined.

Theorem (Gale/Stewart, 1953)

Assuming AC there is a set of reals which is not determined.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 5 / 29

The Projective Hierarchy

Let B be the collection of all Borel sets of reals. Then we define the projective hierarchy as follows. Σ1

1 = analytic sets, i.e. projections of Borel sets,

Π1

n = complements of sets in Σ1 n,

Σ1

n+1 = projections of sets in Π1 n.

A set is projective if it is in Σ1

n (or Π1 n) for some n.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 6 / 29

Determinacy for Different Sets of Reals

Theorem (Martin, 1975)

Assume ZFC. Then every Borel set of reals is determined.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 7 / 29

Determinacy for Different Sets of Reals

Theorem (Martin, 1975)

Assume ZFC. Then every Borel set of reals is determined. Determinacy for all projective sets of reals is not provable in ZFC alone.

Theorem (Martin/Steel, 1985)

Assume ZFC and there are n Woodin cardinals with a measurable cardinal above them all. Then every Σ1

n+1 set is determined.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 7 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 8 / 29

Inner Model Theory

The main goal of inner model theory is to construct L-like models, which we call mice, for stronger and stronger large cardinals.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 9 / 29

G¨

• del’s constructible universe L

Definition

Let E be a set or a proper class. Let J0[E] = ∅ Jα+1[E] = rudE(Jα[E] ∪ {Jα[E]}) Jλ[E] =

• α<λ

Jα[E] for limit λ L[E] =

• α∈Ord

Jα[E] Note that rudE denotes the closure under functions which are rudimentary in E (i.e. basic set operations like minus, union and pairing or intersection with E).

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 10 / 29

Basic properties of L

Condensation Let α be an infinite ordinal and let M ≺ (Lα, ∈). Then the transitive collapse of M is equal to Lβ for some

• rdinal β ≤ α.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 11 / 29

Basic properties of L

Condensation Let α be an infinite ordinal and let M ≺ (Lα, ∈). Then the transitive collapse of M is equal to Lβ for some

• rdinal β ≤ α.

Comparison Let Lα and Lβ for ordinals α and β be initial segments of L. Then one is an initial segment of the other, that means Lα Lβ or Lβ Lα.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 11 / 29

Basic Concepts of Inner Model Theory

Definition

Let M be a countable model of set theory, κ a cardinal and U a κ-complete, nonprincipal ultrafilter on M. Then there is a transitive model N = Ult(M, U) and an elementary embedding iU : M → N with critical point κ. We call N the ultrapower of M via U.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory

Definition

Let M be a countable model of set theory, κ a cardinal and U a κ-complete, nonprincipal ultrafilter on M. Then there is a transitive model N = Ult(M, U) and an elementary embedding iU : M → N with critical point κ. We call N the ultrapower of M via U.

M

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory

Definition

Let M be a countable model of set theory, κ a cardinal and U a κ-complete, nonprincipal ultrafilter on M. Then there is a transitive model N = Ult(M, U) and an elementary embedding iU : M → N with critical point κ. We call N the ultrapower of M via U.

M κ

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory

Definition

Let M be a countable model of set theory, κ a cardinal and U a κ-complete, nonprincipal ultrafilter on M. Then there is a transitive model N = Ult(M, U) and an elementary embedding iU : M → N with critical point κ. We call N the ultrapower of M via U.

M κ N = Ult(M, U)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory

Definition

Let M be a countable model of set theory, κ a cardinal and U a κ-complete, nonprincipal ultrafilter on M. Then there is a transitive model N = Ult(M, U) and an elementary embedding iU : M → N with critical point κ. We call N the ultrapower of M via U.

M κ N = Ult(M, U) iU(κ) iU

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory

Mitchell and Jensen generalized the concept of measures to extenders to

• btain stronger ultrapowers.

Definition

Let M be a countable model of set theory. An extender over M is a system of ultrafilters whose ultrapowers form a directed system, such that they give rise to a single elementary embedding. In fact for every embedding j : M → N there is an extender E over M which gives rise to this embedding.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 13 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N Ult(M, E)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N Ult(M, E)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N Ult(M, E)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison

One key concept of inner model theory is building iterated ultrapowers to compare two models.

M E N Ult(M, E)

⊲

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps.

M0

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps.

M0 Mβ

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps.

M0 Mβ Mα ∋ Eα

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps.

M0 Mβ Mα ∋ Eα Mα+1 ≈ Ult(Mβ,Eα)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees

If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps.

M0 Mβ Mα ∋ Eα Mα+1 ≈ Ult(Mβ,Eα)

The central problem is to choose a cofinal branch such that the direct limit is well-founded.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

The iteration game

More precisely we consider the following two player game G(M, ω1) of length < ω1 for a premouse M.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 16 / 29

The iteration game

More precisely we consider the following two player game G(M, ω1) of length < ω1 for a premouse M. Player I plays at the successor steps: Choose an extender and apply it to the earliest model possible.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 16 / 29

The iteration game

More precisely we consider the following two player game G(M, ω1) of length < ω1 for a premouse M. Player I plays at the successor steps: Choose an extender and apply it to the earliest model possible. Player II plays at the limit steps: Choose a cofinal well-founded branch of the iteration tree constructed so far.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 16 / 29

The iteration game

More precisely we consider the following two player game G(M, ω1) of length < ω1 for a premouse M. Player I plays at the successor steps: Choose an extender and apply it to the earliest model possible. Player II plays at the limit steps: Choose a cofinal well-founded branch of the iteration tree constructed so far. Then player II wins if all the models constructed in the game are well-founded.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 16 / 29

The iteration game

More precisely we consider the following two player game G(M, ω1) of length < ω1 for a premouse M. Player I plays at the successor steps: Choose an extender and apply it to the earliest model possible. Player II plays at the limit steps: Choose a cofinal well-founded branch of the iteration tree constructed so far. Then player II wins if all the models constructed in the game are well-founded.

Definition

We say a premouse M is ω1-iterable iff player II has a winning strategy in the game G(M, ω1). This winning strategy is called an iteration strategy for M.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 16 / 29

Iterable Models and Projective Determinacy

Theorem (Neeman, Woodin)

Let n ≥ 1. Then the following are equivalent. (a) Σ1

n+1-determinacy.

(b) For every x ∈ R the ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) exists.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 17 / 29

Iterable Models and Projective Determinacy

Theorem (Neeman, Woodin)

Let n ≥ 1. Then the following are equivalent. (a) Σ1

n+1-determinacy.

(b) For every x ∈ R the ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) exists.

x M #

n (x) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 17 / 29

Iterable Models and Projective Determinacy

Theorem (Neeman, Woodin)

Let n ≥ 1. Then the following are equivalent. (a) Σ1

n+1-determinacy.

(b) For every x ∈ R the ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) exists.

x M #

n (x)

δ0 δ1 . . . δn−1

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 17 / 29

Iterable Models and Projective Determinacy

Theorem (Neeman, Woodin)

Let n ≥ 1. Then the following are equivalent. (a) Σ1

n+1-determinacy.

(b) For every x ∈ R the ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) exists.

x M #

n (x)

δ0 δ1 . . . δn−1

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 17 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 18 / 29

The L(R)-hierarchy

We define a hierarchy in L(R) as follows. J1(R) = Vω+1 Jα+1(R) = rud(Jα(R) ∪ {Jα(R)}) Jλ(R) =

• α<λ

Jα(R) for limit λ L(R) =

• α∈Ord

Jα(R)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 19 / 29

The L(R)-hierarchy extends the projective hierarchy

We can consider a hierarchy of pointclasses like ΣJβ(R)

n

along the model L(R).

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 20 / 29

The L(R)-hierarchy extends the projective hierarchy

We can consider a hierarchy of pointclasses like ΣJβ(R)

n

along the model L(R). Recall that J1(R) = Vω+1. We in fact have that ΣJ1(R)

n

∩ P(R) = Σ1

n,

so the L(R)-hierarchy extends the projective hierarchy.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 20 / 29

Pointclasses beyond the projective hierarchy

In some sense this hierarchy looks like infinitely many copies of the projective hierarchy, but between these copies we might have different forms of gaps.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 21 / 29

Pointclasses beyond the projective hierarchy

In some sense this hierarchy looks like infinitely many copies of the projective hierarchy, but between these copies we might have different forms of gaps.

Definition

We say [α, β] is a Σ1-gap iff (i) Jα(R) ≺Σ1 Jβ(R), (ii) for all α′ < α, Jα′(R) ⊀Σ1 Jα(R), and (iii) for all β′ > β, Jβ(R) ⊀Σ1 Jβ′(R).

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 21 / 29

How do the corresponding mice look like?

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 22 / 29

How do the corresponding mice look like?

At the projective levels we consider mice with finitely many Woodin cardinals.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 22 / 29

How do the corresponding mice look like?

At the projective levels we consider mice with finitely many Woodin cardinals. So maybe for the L(R)-hierarchy we have to consider mice with infinitely many Woodin cardinals.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 22 / 29

How do the corresponding mice look like?

At the projective levels we consider mice with finitely many Woodin cardinals. So maybe for the L(R)-hierarchy we have to consider mice with infinitely many Woodin cardinals. No!

Theorem (Woodin)

The following are equiconsistent.

(1) There exist infinitely many Woodin cardinals. (2) L(R)-determinacy.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 22 / 29

Mice beyond the projective hierarchy

The mice we want to construct from determinacy at these levels also have finitely many Woodin cardinals. What gives them strength here is that we want them to capture certain sets of reals.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

The mice we want to construct from determinacy at these levels also have finitely many Woodin cardinals. What gives them strength here is that we want them to capture certain sets of reals.

Definition

Let A ⊂ R, let M be a countable mouse and δ an uncountable cardinal of

• M. Then we say M captures A at δ iff there is an M-term τ for A at δ

and M absorbs reals at δ.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

The mice we want to construct from determinacy at these levels also have finitely many Woodin cardinals. What gives them strength here is that we want them to capture certain sets of reals.

Definition

Let A ⊂ R, let M be a countable mouse and δ an uncountable cardinal of

• M. Then we say M captures A at δ iff there is an M-term τ for A at δ

and M absorbs reals at δ.

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A M

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A M δ

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A M δ τ Col(ω,δ)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A M δ τ P i

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A M δ τ P i i(δ)

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse, δ an uncountable cardinal of M, and let τ ∈ MCol(ω,δ) be a term. We say τ is an M-term for A at δ iff whenever P is an iterate of M via i : M → P then whenever g is Col(ω, i(δ))-generic over P, then A ∩ P[g] = i(τ)g.

A M δ τ P i i(δ) Col(ω,i(δ))

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 23 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse and δ an uncountable cardinal of

• M. Then we say M captures A at δ iff there is an M-term τ for A at δ

and M absorbs reals at δ.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let A ⊂ R, let M be a countable mouse and δ an uncountable cardinal of

• M. Then we say M captures A at δ iff there is an M-term τ for A at δ

and M absorbs reals at δ.

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M δ

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M δ η

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M δ η x

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M δ η x M∗ i(δ) i(η) i

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M δ η x M∗ i(δ) i(η) i M∗∗ j(i(δ)) i(η) j

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Mice beyond the projective hierarchy

Definition

Let M be a countable mouse and let δ be a cardinal in M. Then we say that M absorbs reals at δ iff for every ordinal η < δ and for every real x, whenever i : M → M∗ is an iteration based on M|η, then there exists an iteration j : M∗ → M∗∗ based on M∗|i(δ) above i(η) such that x ∈ M∗∗[g], for some Col(ω, j(i(δ)))-generic g over M∗∗.

M δ η x M∗ i(δ) i(η) i M∗∗ j(i(δ)) i(η) j Col(ω,j(i(δ)))

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 24 / 29

Examples of mice capturing sets of reals

Lemma

The ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) captures Σ1 n+1-sets of reals at its bottom Woodin cardinal.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 25 / 29

Examples of mice capturing sets of reals

Lemma

The ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) captures Σ1 n+1-sets of reals at its bottom Woodin cardinal.

Lemma

Let A =

• k∈ω

Ak be a set of reals. Moreover let N be a countable mouse with a cardinal δ such that N captures every set Ak at δ for each k < ω.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 25 / 29

Examples of mice capturing sets of reals

Lemma

The ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) captures Σ1 n+1-sets of reals at its bottom Woodin cardinal.

Lemma

Let A =

• k∈ω

Ak be a set of reals. Moreover let N be a countable mouse with a cardinal δ such that N captures every set Ak at δ for each k < ω. Then a mouse M which has a Woodin cardinal, contains N and knows how to iterate N captures the set A at its Woodin cardinal.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 25 / 29

Examples of mice capturing sets of reals

Lemma

The ω1-iterable countable model of set theory with n Woodin cardinals M#

n (x) captures Σ1 n+1-sets of reals at its bottom Woodin cardinal.

Lemma

Let A =

• k∈ω

Ak be a set of reals. Moreover let N be a countable mouse with a cardinal δ such that N captures every set Ak at δ for each k < ω. Then a mouse M which has a Woodin cardinal, contains N and knows how to iterate N captures the set A at its Woodin cardinal.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 25 / 29

Hybrid mice

To obtain models which capture certain sets of reals, in general ordinary mice do not seem to be enough. We will construct hybrid mice, i.e. mice which know how to iterate another mouse.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 26 / 29

Hybrid mice

To obtain models which capture certain sets of reals, in general ordinary mice do not seem to be enough. We will construct hybrid mice, i.e. mice which know how to iterate another mouse.

Definition

Let N be a countable mouse and Σ an iteration strategy for N. We say M is a (hybrid) Σ-mouse iff M is a mouse build over N where the iteration strategy Σ is added during the construction.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 26 / 29

Hybrid mice

To obtain models which capture certain sets of reals, in general ordinary mice do not seem to be enough. We will construct hybrid mice, i.e. mice which know how to iterate another mouse.

Definition

Let N be a countable mouse and Σ an iteration strategy for N. We say M is a (hybrid) Σ-mouse iff M is a mouse build over N where the iteration strategy Σ is added during the construction.

N M δ

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 26 / 29

Hybrid mice

To obtain models which capture certain sets of reals, in general ordinary mice do not seem to be enough. We will construct hybrid mice, i.e. mice which know how to iterate another mouse.

Definition

Let N be a countable mouse and Σ an iteration strategy for N. We say M is a (hybrid) Σ-mouse iff M is a mouse build over N where the iteration strategy Σ is added during the construction.

N M δ Σ

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 26 / 29

Hybrid mice

To obtain models which capture certain sets of reals, in general ordinary mice do not seem to be enough. We will construct hybrid mice, i.e. mice which know how to iterate another mouse.

Definition

Let N be a countable mouse and Σ an iteration strategy for N. We say M is a (hybrid) Σ-mouse iff M is a mouse build over N where the iteration strategy Σ is added during the construction. Note that we need an iteration strategy Σ that behaves nicely and we have to be careful during the construction to make sure that we obtain a reasonable model.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 26 / 29

Hybrid mice from determinacy

Theorem

Let α < β be ordinals such that [α, β] is a weak Σ1-gap, let k ≥ 0, and let A ∈ Γ = Σn(Jβ(R)) ∩ P(R), where n < ω is the least natural number such that ρn(Jβ(R)) = R. Moreover assume that every Π1

2k+5Γ-definable set of reals is determined.

Then there exists an ω1-iterable hybrid Σ-premouse N which captures every set of reals in the pointclass Σ1

k(A) or Π1 k(A).

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 27 / 29

Hybrid mice from determinacy

Theorem

Let α < β be ordinals such that [α, β] is a weak Σ1-gap, let k ≥ 0, and let A ∈ Γ = Σn(Jβ(R)) ∩ P(R), where n < ω is the least natural number such that ρn(Jβ(R)) = R. Moreover assume that every Π1

2k+5Γ-definable set of reals is determined.

Then there exists an ω1-iterable hybrid Σ-premouse N which captures every set of reals in the pointclass Σ1

k(A) or Π1 k(A).

The mouse M we construct is in fact of the form MΣ,#

k

(N) for some countable mouse N.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 27 / 29

Open questions

Can we reduce the hypothesis in this theorem to determinacy for Π1

k+2Γ-definable sets of reals?

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 28 / 29

Open questions

Can we reduce the hypothesis in this theorem to determinacy for Π1

k+2Γ-definable sets of reals?

What is the “optimal” mouse to consider for example at the level ∆J2(R)

2

(the first interesting level after the projective hierarchy)?

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 28 / 29

Open questions

Can we reduce the hypothesis in this theorem to determinacy for Π1

k+2Γ-definable sets of reals?

What is the “optimal” mouse to consider for example at the level ∆J2(R)

2

(the first interesting level after the projective hierarchy)?

Conjecture (Steel, Woodin, 2015)

Let P be the minimal ladder mouse. Then for all reals x, we have that x ∈ P ∩ R iff x is ∆J2(R)

2

• definable in a countable ordinal.

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 28 / 29

Thank you for your attention!

For reference see “Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy” (Dissertation), soon available at http://boolesrings.org/sandrauhlenbrock/publications/dissertation/

Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 29 / 29