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Partition Properties for Non-Ordinal Sets Under the Axiom of - - PowerPoint PPT Presentation

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Background J onsson Properties for R J onsson Properties for General Sets J onsson Properties for R / E 0 Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy Jared Holshouser and Stephen Jackson University of North

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

Jared Holshouser and Stephen Jackson

University of North Texas

BEST 2016

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Finite Partition Properties

Definition

For κ a cardinal and n ∈ ω, [κ]n = {(α1, · · · , αn) ∈ κn : α1 < · · · < αn}. We also set [κ]<ω =

n∈ω[κ]n.

Definition

Let κ, λ, δ be cardinals and n ∈ ω.

◮ [κ]<ω λ

→ [κ]<ω

δ

means: for every f : [κ]<ω → λ, there is an H ⊆ κ so that |H| = κ and |f [[H]n]| ≤ δ for all n.

◮ [κ]<ω <λ → [κ]<ω δ

means: [κ]<ω

µ

→ [κ]<ω

δ

for all µ < λ.

◮ κ is Ramsey iff [κ]<ω 2

→ [κ]<ω

1 . ◮ κ is Rowbottom iff [κ]<ω <κ → [κ]<ω ω .

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Finite Partition Properties

Definition

κ is J´

  • nsson iff for every f : [κ]<ω → κ, there is an H ⊆ κ so that

|H| = κ and f [[H]<ω] = κ.

Remark

In, ZFC, Ramsey implies Rowbottom and J´

  • nsson, and both

Rowbottom and J´

  • nsson imply the existence of 0# and thus that

V = L.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Some Determinacy Notions

Definition

Recall that under the axiom of determinacy (AD), R cannot be well-ordered. We define Θ to be least cardinal that R does not surject onto.

Definition

Recall that L(R) is the minimal universe of ZF which contains R. Under large cardinal hypotheses, L(R) is a model of AD, and its theory is absolute for very complex statements.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Some Determinacy Notions

Definition

Recall that under the axiom of determinacy (AD), R cannot be well-ordered. We define Θ to be least cardinal that R does not surject onto.

Definition

Recall that L(R) is the minimal universe of ZF which contains R. Under large cardinal hypotheses, L(R) is a model of AD, and its theory is absolute for very complex statements.

Remark

It has been shown that under AD, ordinary cardinals have large cardinal properties in L(R). For instance, ω1 is a measurable cardinal.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Finite Partition Properties Under AD

In 2015, S. Jackson, R. Ketchersid, F. Schlutzenberg, and W.H. Woodin [3] proved the following:

Theorem (AD + V = L(R), J/K/S/W)

Let κ < Θ be an uncountable cardinal. Then:

  • 1. If cf(κ) = ω, then κ is Rowbottom.
  • 2. κ is J´
  • nsson. In fact, if λ is a cardinal between ω1 and κ, and

f : [κ]<ω → λ, then there is an H ⊆ κ so that |H| = κ and |λ − f [[H]<ω]| = λ.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Finite Partition Properties Under AD

In 2015, S. Jackson, R. Ketchersid, F. Schlutzenberg, and W.H. Woodin [3] proved the following:

Theorem (AD + V = L(R), J/K/S/W)

Let κ < Θ be an uncountable cardinal. Then:

  • 1. If cf(κ) = ω, then κ is Rowbottom.
  • 2. κ is J´
  • nsson. In fact, if λ is a cardinal between ω1 and κ, and

f : [κ]<ω → λ, then there is an H ⊆ κ so that |H| = κ and |λ − f [[H]<ω]| = λ. In this paper, they asked whether or not there were non-ordinal J´

  • nsson cardinals. In particular, is R J´
  • nsson?

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Reframing the Question

Definition

For any set A, [A]n = {s ⊆ X : |s| = n} and [A]<ω =

n∈ω[A]n.

Definition

Let A and B be infinite sets.

◮ (A, B) is Ramsey iff for any f : [A]<ω → B, there is an X ⊆ A so that

|X| = |A| and f is constant on each [X]n.

◮ (A, B) is Rowbottom iff for any f : [A]<ω → B, there is an X ⊆ A so

that |X| = |A| and f [[X]<ω] is countable.

◮ (A, B) is a strong J´

  • nsson pair iff for any f : [A]<ω → B, there is an

X ⊆ A so that |X| = |A| and |B − f [[X]<ω]| = |B|.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Tools From Descriptive Set Theory

We use the following repeatedly.

Lemma (Fusion Lemma)

For each s ∈ 2<ω let Ps be a perfect set so that

  • 1. lim|s|→∞ diam(Ps) = 0, and
  • 2. for all s ∈ 2<ω, Ps0 ∩ Ps1 = ∅ and Ps0, Ps1 ⊆ Ps.

Then the fusion P =

f ∈2ω

  • n∈ω Pf |n of Ps : s ∈ 2<ω is a

perfect set.

Theorem (Mycielski)

Suppose Cn ⊆ (2ω)n are comeager for all n ∈ ω. Then there is a perfect set P ⊆ 2ω so that [P]n ⊆ Cn for all n.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

R is Strongly J´

  • nsson

Theorem (AD, Holshouser/Jackson)

R is Strongly J´

  • nsson.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

R is Strongly J´

  • nsson

Theorem (AD, Holshouser/Jackson)

R is Strongly J´

  • nsson.

Proof.

◮ We can break f into component functions, fn. ◮ Find comeager sets on which the fn are continuous. ◮ Use the result of Mycielski[4] to thread a perfect set through

the comeager sets.

◮ Use continuity and the fusion lemma to inductively thin out

the range of the fn.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Domain f1 Range

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Domain f1 Range

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Domain f1 Range

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Domain f1 Range

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Domain f1 Range

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

R and Cardinals

Proposition (AD)

If κ < Θ is an uncountable cardinal, then (R, κ) and (κ, R) are Rowbottom.

Proposition (AD + V = L(R), Holshouser/Jackson)

Let κ, λ < Θ be uncountable cardinals. Suppose A, B ∈ {κ, λ, R, κ ∪ R, κ × R, λ ∪ R, λ × R} Then (A, B) is a strong J´

  • nsson pair.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

R and Cardinals

Proposition (AD)

If κ < Θ is an uncountable cardinal, then (R, κ) and (κ, R) are Rowbottom.

Proposition (AD + V = L(R), Holshouser/Jackson)

Let κ, λ < Θ be uncountable cardinals. Suppose A, B ∈ {κ, λ, R, κ ∪ R, κ × R, λ ∪ R, λ × R} Then (A, B) is a strong J´

  • nsson pair.

What about other non-ordinal sets?

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Describing More General Sets

Suppose X ∈ LΘ(R). Then there is a surjection F : R → X. We can define an equivalence relation E on R by xEy ⇐ ⇒ F(x) = F(y). Note that X is in bijection with R/E. So we only need to consider quotients of R.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

J´

  • nsson Properties for General Quotients

There is a (possibly not unique) decomposition of R/E into a well-ordered component and another component which R surjects onto and injects into [2]. Call the surjection φX and the injection φX. Either of these components could be empty.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

J´

  • nsson Properties for General Quotients

There is a (possibly not unique) decomposition of R/E into a well-ordered component and another component which R surjects onto and injects into [2]. Call the surjection φX and the injection φX. Either of these components could be empty.

Theorem (AD + V = L(R), Holshouser/Jackson)

Suppose that X ∈ LΘ(R) is in bijection with κ ∪ A, where κ is an uncountable cardinal and R maps onto and into A. Similarly, suppose Y ∈ LΘ(R) is in bijection with λ ∪ B. Let f : [κ ∪ A]<ω → λ ∪ B. Then there are perfect P, Q ⊆ R and there is an H ⊆ κ with |H| = κ so that |λ − f [[H ∪ φA[P]]<ω]| = λ and f [[H ∪ φA[P]]<ω] ∩ φB[Q] = ∅.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

J´

  • nsson Properties for General Quotients

There is a (possibly not unique) decomposition of R/E into a well-ordered component and another component which R surjects onto and injects into [2]. Call the surjection φX and the injection φX. Either of these components could be empty.

Theorem (AD + V = L(R), Holshouser/Jackson)

Suppose that X ∈ LΘ(R) is in bijection with κ ∪ A, where κ is an uncountable cardinal and R maps onto and into A. Similarly, suppose Y ∈ LΘ(R) is in bijection with λ ∪ B. Let f : [κ ∪ A]<ω → λ ∪ B. Then there are perfect P, Q ⊆ R and there is an H ⊆ κ with |H| = κ so that |λ − f [[H ∪ φA[P]]<ω]| = λ and f [[H ∪ φA[P]]<ω] ∩ φB[Q] = ∅. This is unsatisfactory as this result does not give us bijections.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Background for E0

Recall the following:

Definition

Let x, y ∈ 2ω. Then xE0y iff (∃N)(∀n ≥ N)[x(n) = y(n)]. Note that 2ω/E0 has no definable linear ordering and E0 has no definable transversal.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Background for E0

Recall the following:

Definition

Let x, y ∈ 2ω. Then xE0y iff (∃N)(∀n ≥ N)[x(n) = y(n)]. Note that 2ω/E0 has no definable linear ordering and E0 has no definable transversal. The following is a corollary of the Glimm-Effros Dichotomy [1]:

Corollary (AD)

Suppose H ⊆ 2ω/E0. Then H satisfies exactly one of the following:

◮ H is countable, ◮ H is in bijection with R, or ◮ H is in bijection with 2ω/E0.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Mycielski for E0

Definition

A ⊆ 2ω has power E0 iff A is E0-saturated and A/E0 is in bijection with 2ω/E0.

Definition

For n ∈ ω and A ⊆ 2ω, let [A]n

E0 = {

x ∈ [A]n : |{[x1]E0, · · · , [xn]E0}| = n}

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Mycielski for E0

Definition

A ⊆ 2ω has power E0 iff A is E0-saturated and A/E0 is in bijection with 2ω/E0.

Definition

For n ∈ ω and A ⊆ 2ω, let [A]n

E0 = {

x ∈ [A]n : |{[x1]E0, · · · , [xn]E0}| = n} We were able to prove the following Mycielski style result.

Theorem (Holshouser/Jackson)

Suppose that Cn ⊆ (2ω)n are comeager and E0-saturated for all n ∈ ω. Then there is an A ⊆ 2ω of power E0 so that [A]n

E0 ⊆ Cn

for all n.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

R/E0 is Strongly J´

  • nsson

Theorem (AD, Holshouser/Jackson)

2ω/E0 is strongly J´

  • nsson.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

R/E0 is Strongly J´

  • nsson

Theorem (AD, Holshouser/Jackson)

2ω/E0 is strongly J´

  • nsson.

Proof.

◮ We can lift f : [2ω/E0]<ω → 2ω/E0 to a function F : [2ω]<ω → 2ω so that

  • aE0

b ⇐ ⇒ F( a) ∈ f ({[b1]E0, · · · , [bn]E0}).

◮ We can break F into component functions, Fn. ◮ Find comeager sets on which the Fn are continuous. ◮ Use the Mycielski-style result for E0 to thread a power E0 set through the

comeager sets.

◮ Use continuity and the techniques of the Mycielski-style result to

inductively thin out the range of the Fn.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Combinations

Proposition (AD + V = L(R), Holsouser/Jackson)

Let κ, λ < Θ be uncountable cardinals. Suppose A, B ∈ {κ, λ, R, 2ω/E0, κ ∪ R, κ × R, λ ∪ R, λ × R κ ∪ 2ω/E0, κ × 2ω/E0, λ ∪ 2ω/E0, λ × 2ω/E0} Then (A, B) is a strong J´

  • nsson pair.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

More Finite Partition Properties

Proposition (AD, Holshouser/Jackson)

Let κ < Θ be an uncountable cardinal. Then

◮ (2ω/E0, R) is Ramsey, ◮ (2ω/E0, κ) is Ramsey, and ◮ (κ, 2ω/E0) is Rowbottom.

Proposition (AD + V = L(R), Holshouser/Jackson)

Suppose λ, κ < Θ are uncountable cardinals. Then

◮ (κ ∪ 2ω/E0, R) is Rowbottom, ◮ if cf(κ) = ω and λ < κ, then (κ ∪ 2ω/E0, λ) is Rowbottom,

and

◮ (2ω/E0, κ ∪ R) and (2ω/E0, κ × R) are Ramsey.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Further Work

◮ Can the result be extended to well-ordered unions of

hyperfinite quotients of R?

◮ Do infinite partition properties hold for 2ω/E0? ◮ Can we get this Mycielski style result for other equivalence

relations?

◮ Can the full J´

  • nsson result be proved for general equivalence

relations?

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

Thanks For Listening!

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

References I

[1] S. Gao. Invariant Descriptive Set Theory. CRC Press, 2008. [2] L. Harrington and S. Shelah. Counting equivalence classes for co-κ-suslin equivalence relations. Logic Colloquium, 108, 1980. [3] S. Jackson, R. Ketchersid, F. Schlutzenberg, and W. Woodin. Ad and J´

  • nsson cardinals in L(R).

Journal of Symbolic Logic, 79, 2014.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy

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Background J´

  • nsson Properties for R

J´

  • nsson Properties for General Sets

J´

  • nsson Properties for R/E0

References II

[4] J. Mycielski. Independent sets in topological algebras.

  • Fund. Math., 55, 1964.

Jared Holshouser and Stephen Jackson University of North Texas Partition Properties for Non-Ordinal Sets Under the Axiom of Determinacy