victoria gitman · 2020. 10. 10. · victoria gitman elementary embeddings and smaller large...

Elementary embeddings and smaller large cardinals

Victoria Gitman

[email protected]://victoriagitman.github.io

Oxford set theory seminar

April 28, 2020

Victoria Gitman Elementary embeddings and smaller large cardinals Oxford 1 / 29

Large elementary embeddings

Elementary embeddings and larger large cardinals

A common theme in the definitions of larger large cardinals is the existence of elementaryembeddings from the universe V into some inner model M.

A cardinal κ is measurable if there exists an elementary embedding j : V → M withcrit(j) = κ.

A cardinal κ is strong if for every λ > κ, there is an elementary embeddingj : V → M with crit(j) = κ and Vλ ⊆ M.

A cardinal κ is supercompact if for every λ > κ, there is an elementary embeddingj : V → M with crit(j) = κ and Mλ ⊆ M.

The closer M is to V the stronger the large cardinal.



Elementary embeddings and ultrafilters

Suppose κ is a cardinal and U ⊆ P(κ) is an ultrafilter.

U is α-complete, for a cardinal α, if whenever β < α and {Aξ | ξ < β} is a sequenceof sets such that Aξ ∈ U, then

⋂ξ<β Aξ ∈ U.

U is normal if whenever {Aξ | ξ < κ} is a sequence of sets such that Aξ ∈ U, thenthe diagonal intersection ∆ξ<κAξ ∈ U. ∆ξ<κAξ = {α < κ | α ∈

⋂ξ<α Aξ}

Theorem: The ultrapower of V by U is well-founded if and only if U is an ω1-complete.

Observations:

If U is normal and all the tails sets κ \ α ∈ U for α < κ, then U is κ-complete.

If U is ω1-complete, then we get an elementary embedding jU : V → M, where M isthe Mostowski collapse of the ultrapower.

If U is κ-complete, then jU : V → M has critical point κ.

Proposition: Suppose j : V → M is an elementary embedding with crit(j) = κ. ThenU = {A ⊆ κ | κ ∈ j(A)} is a normal ultrafilter.

We call U the ultrafilter generated by κ via j .



Iterated ultrapowers

Suppose κ is a cardinal and U ⊆ P(κ) is an ultrafilter.

The ultrapower construction with U can be iterated as follows.

Let V = M0 and j01 : M0 → M1 be the ultrapower of V by U.

Let j12 : M1 → M2 be the ultrapower of M1 by j01(U), which is an ultrafilter onj01(κ) in M1.

Let j12 ◦ j01 = j0,2 : M0 → M2.

Inductively, given jξγ : Mξ → Mγ for ξ < γ < δ, define:

if δ = α + 1, let jα,δ : Mα → Mδ be the ultrapower of Mα by j0α(U).

if δ is a limit, let Mδ be the direct limit the system of iterated ultrapowerembeddings constructed so far.

Theorem: (Gaifman) If U is ω1-complete, then the iterated ultrapowers Mξ for ξ ∈ Ordare well-founded.

If Mξ is well-founded, then Mξ+1 is well-founded, since j0ξ(U) is ω1-complete in Mξ.

It suffices to see that the countable limit stages Mξ for ξ < ω1 are well-founded.


Small elementary embeddings

Smaller large cardinals

Definition: A cardinal κ is weakly compact if every coloring f : [κ]2 → 2 of pairs ofelements of κ in 2 colors has a homogeneous set of size κ.

Theorem: The following are equivalent:

κ is weakly compact.

(Erdos, Tarski) κ is inaccessible and the tree property holds at κ.

(Kiesler, Tarski) Every <κ-satisfiable theory of size κ in Lκ,κ is satisfiable.

Definition: A cardinal κ is ineffable if for every sequence {Aξ | ξ < κ}with Aξ ⊆ ξ, there is a A ⊆ κ and a stationary set S such that for allξ ∈ S , A ∩ ξ = Aξ.

Theorem: (Kunen, Jensen) A cardinal κ is ineffable if and only if everycoloring f : [κ]2 → 2 of pairs of elements of κ in 2 colors has astationary homogeneous set.

Definition:

A cardinal κ is α-Erdos if every coloring f : [κ]<ω → 2 of finitetuples of elements of κ in 2 colors has a homogeneous set oforder-type α.

A cardinal κ is Ramsey if κ is κ-Erdos.

weakly compact

ineffable

α-Erdos (α ∈ ω1)

ω1-Erdos

Ramsey

L



Weak κ-models

Smaller large cardinals κ usually imply existence of elementary embeddings of models of(weak) set theory of size κ.

Suppose κ is a cardinal.

Definition:

A weak κ-model is a transitive model M |= ZFC− of size κ with κ ∈ M.ZFC− is the theory ZFC without the powerset axiom with the collection scheme instead of the replacement scheme.

A κ-model M is a weak κ-model such that M<κ ⊆ M.This is the maximum possible closure for a model of size κ.

A weak κ-model is simple if κ is the largest cardinal of M.

Natural simple weak κ-models arise as elementary substructures of Hκ+ . Hθ = {x | |TCl(x)| < θ}

Observations:

If M ≺ Hκ+ has size κ and κ ⊆ M, then M is a simple weak κ-model.

If κ is inaccessible, then there are simple κ-models M ≺ Hκ+ .



Small ultrafilters and elementary embeddings

Suppose M is a weak κ-model.

Let PM(κ) = {A ⊆ κ | A ∈ M}. PM (κ) typically won’t be an element of M.

Definition: A set U ⊆ PM(κ) is an M-ultrafilter if it contains the tail sets κ \ α and thestructure

〈M,∈,U〉 |= “U is a normal ultrafilter on κ.”

U is an ultrafilter measuring PM(κ).

U is closed under diagonal intersections ∆ξ<κAξ for sequences {Aξ | ξ < κ} ∈ M.

Typically, U /∈ M.

Typically, separation and collection will fail badly in the structure 〈M,∈,U〉.We will see why later on.

Definition: Suppose U is an M-ultrafilter.

U is α-complete, for a cardinal α, if whenever β < α and {Aξ | ξ < β} is a sequenceof sets such that Aξ ∈ U, then

⋂ξ<β Aξ 6= ∅.

U is good if the ultrapower of M by U is well-founded.




Suppose M is a weak κ-model and U is an M-ultrafilter.

Observations:

If U is ω1-complete, then U is good.We will see shortly that the converse fails.

If M is a κ-model, then U is ω1-complete.

Proposition:

If U is a good M-ultrafilter, then the Mostowski collapse of the ultrapower yields anelementary embedding jU : M → N with crit(jU) = κ.

Suppose j : M → N is an elementary embedding with crit(j) = κ.

Then U = {A ∈ M | A ⊆ κ and κ ∈ j(A)} is a good M-ultrafilter.

We call U the M-ultrafilter generated by κ via j .



Iterating small ultrapowers

Suppose M is a weak κ-model, U is an M-ultrafilter, and jU : M → N is the ultrapowerembedding.

To iterate the ultrapower construction, we need to define “jU(U)”.

Definition: An M-ultrafilter U is weakly amenable if for every A ∈ M with |A|M ≤ κ,U ∩ A ∈ M.

If M is simple, then U is fully amenable.

jU(U) = {A ⊆ j(κ) | A = [f ] and {ξ < κ | f (ξ) ∈ U} ∈ U}.

Weakly amenable M-ultrafilters U are “partially internal to M”.



Weakly amenable M-ultrafilters

Suppose M is a weak κ-model and U is an M-ultrafilter.

Proposition: U is weakly amenable if and only if 〈M,∈,U〉 satisfies Σ0-separation.

Definition: An elementary embedding j : M → N with crit(j) = κ is κ-powersetpreserving if PM(κ) = PN(κ).

Proposition:

If U is good and weakly amenable, then the ultrapower jU : M → N is κ-powersetpreserving.

I If M is simple, then M = HNκ+ .

If j : M → N is κ-powerset preserving, then U, the M-ultrafilter generated by κ viaj , is weakly amenable.

In an ultrapower jU : M → N by a weakly amenable M-ultrafilter, κ-powersetpreservation creates reflection between M and its ultrapower N.



Elementary embedding characterizations of weakly compact cardinals

Theorem: The following are equivalent for an inaccessible cardinal κ.

κ is weakly compact.

For every A ⊆ κ, there is a weak κ-model M, with A ∈ M, for which there is a goodM-ultrafilter.

For every A ⊆ κ, there is a κ-model M, with A ∈ M, for which there is anM-ultrafilter.

For every A ⊆ κ, there is a κ-model M ≺ Hκ+ , with A ∈ M, for which there is anM-ultrafilter.

For every weak κ-model M, there is a good M-ultrafilter.

Question: Can we get weakly amenable M-ultrafilters U?

We will see that the more “internal” the M-ultrafilter U is to M, the stronger theassociated large cardinal.



α-iterable cardinals

Suppose M is a weak κ-model.

Definition: An M-ultrafilter U is α-iterable if it is weakly amenable and has α-manywell-founded iterated ultrapowers. U is iterable if it is α-iterable for every α ∈ Ord.

Proposition: (Gaifman) If an M-ultrafilter U is ω1-iterable, then U is iterable.

Theorem: (Kunen) If an M-ultrafilter U is ω1-complete, then U is iterable.

Definition: (G., Welch) A cardinal κ is α-iterable, for 1 ≤ α ≤ ω1, if for every A ⊆ κthere is a weak κ-model M, with A ∈ M, for which there is an α-iterable M-ultrafilter.

Theorem:

(G.) A 1-iterable cardinal κ is a limit of ineffable cardinals.

(G., Schindler) Suppose λ is additively indecomposable. A λ+ 1-iterable cardinalhas a λ-Erdos cardinal below it. A λ-Erdos cardinal is a limit of λ-iterable cardinals.

(G., Welch) An α-iterable cardinal is a limit of β-iterable cardinals for all β < α.

(G., Welch) If α < ω1, then an α-iterable cardinal is downward absolute to L.



Elementary embedding characterization of Ramsey cardinals

Theorem: (Mitchell) A cardinal κ is Ramsey if and only if for everyA ⊆ κ there is a weak κ-model M, with A ∈ M, for which there is aweakly amenable ω1-complete M-ultrafilter.

Theorem: (Sharpe, Welch) A Ramsey cardinal is a limit of ω1-iterablecardinals.

Question: Can we strengthen the Ramsey embedding characterizationby replacing weak κ-model with κ-model or κ-model elementary inHκ+ , etc.?

Definition:

A cardinal κ is strongly Ramsey if for every A ⊆ κ there is aκ-model M, with A ∈ M, for which there is a weakly amenableM-ultrafilter.

A cardinal κ is super Ramsey if for every A ⊆ κ there is a κ-modelM ≺ Hκ+ , with A ∈ M, for which there is a weakly amenableM-ultrafilter.

weakly compact

α-iterable (α ∈ ω1)

ω1-iterable

Ramsey

L

measurable



Strongly and super Ramsey cardinals

Theorem: (G.)

A measurable cardinal is a limit of super Ramsey cardinals.

A super Ramsey cardinal is a limit of strongly Ramsey cardinals.

A strongly Ramsey cardinal is a limit of Ramsey cardinals.

It is inconsistent for every κ-model to have a weakly amenableM-ultrafilter.

We can weaken strongly Ramsey cardinals to assert that for everyA ⊆ κ there is a weak κ-model M, with A ∈ M, such that Mω ⊆ M forwhich there is a weakly amenable M-ultrafilter. Such a cardinal isalready a limit of Ramsey cardinals.

Question: Can we stratify by closure on the weak κ-model M?

Question: Can we have elementary embeddings on models elementaryin some large Hθ? weakly compact


ω1-iterable

Ramsey

L

strongly Ramsey

super Ramsey

measurable



α-Ramsey cardinalsDefinition:

An imperfect weak κ-model is an ∈-model M |= ZFC− such that κ+ 1 ⊆ M.

An imperfect κ-model is an imperfect weak κ-model M such that M<κ ⊆ M.

Definition: (Holy, Schlicht) A cardinal κ is α-Ramsey for a regular α, with ω1 ≤ α ≤ κ,if for every A ⊆ κ and arbitrarily large regular θ, there is an imperfect weak κ-modelM ≺ Hθ, with A ∈ M, such that M<α ⊆ M for which there is a weakly amenableM-ultrafilter.

Proposition: (Holy, Schlicht) The following are equivalent.

κ is α-Ramsey.

For every A and arbitrarily large regular θ there is an imperfect weak κ-modelM ≺ Hθ, with A ∈ M, such that M<α ⊆ M for which there is a weakly amenableM-ultrafilter.

For arbitrarily large regular θ there is an imperfect weak κ-model M ≺ Hθ such thatM<α ⊆ M for which there is a weakly amenable M-ultrafilter.

Theorem: (Holy, Schlicht)

A measurable cardinal is a limit of κ-Ramsey cardinals κ.

A κ-Ramsey cardinal κ is a limit of super Ramsey cardinals.

(G.) A strongly Ramsey cardinal is a limit of cardinals α which are α-Ramsey.

An ω1-Ramsey cardinal is a limit of Ramsey cardinals.



Games with κ-models and small ultrafilters

Definition: (Holy, Schlicht) Fix regular α and θ such that ω1 ≤ α ≤ κ and θ > κ. Thegame RamseyG θα(κ) is played by the challenger and the judge.

At every stage γ < α:

the challenger plays an imperfect κ-model Mγ ≺ Hθ extending his previous moves,

the judge responds with an Mγ-ultrafilter Uγ extending her previous moves,

{〈Mγ ,∈,Uγ〉 | γ < γ} ∈ Mγ .

The judge wins if she can play for α-many moves and otherwise the challenger wins.

Observations: Suppose the judge wins a run of the game RamseyG θα(κ).

M =⋃γ<αMγ is closed under <α-sequences.

U =⋃γ<α Uγ is a weakly amenable M-ultrafilter.

Definition: The game RamseyG∗θα (κ) is played like RamseyG θα(κ), but now the judge

plays structures 〈Nγ ,∈,Uγ〉 such that Nγ is a κ-model with PMγ (κ) ⊆ Nγ and Uγ is anNγ-ultrafilter.

Question: Why games?

Theorem: (G.) Suppose κ is weakly compact. The property that given a κ-model M, anM-ultrafilter U, and a κ-model M extending M, we can always find a M-ultrafilter Uextending U is inconsistent.



Games and α-Ramsey cardinals

Theorem: (Holy, Schlicht) The existence of a winning strategy foreither player in the games RamseyG θα(κ) or RamseyG∗θ

α (κ) isindependent of θ.

Theorem: (Holy, Schlicht) The following are equivalent.

κ is α-Ramsey.

The challenger doesn’t have a winning strategy in the gameRamseyG θα(κ) for some/all θ.

The challenger doesn’t have a winning strategy in the gameRamseyG∗θ

α (κ) for some/all θ.

For every A ∈ H(2κ)+ , there is an imperfect weak κ-modelM ≺ H(2κ)+ , with A ∈ M, such that M<α ⊆ M for which there isa weakly amenable M-ultrafilter.

Theorem: (Holy, Schlicht) Every β-Ramsey cardinal is a limit ofα-Ramsey cardinals for α < β.

weakly compact

α-iterable

ω1-iterable

Ramsey

L

α-Ramsey

strongly Ramsey

super Ramsey

κ-Ramsey

measurable



The structure 〈M,∈,U〉Suppose κ is inaccessible, M is a simple weak κ-model, with Vκ ∈ M and U is a weaklyamenable M-ultrafilter.

Proposition: The structure 〈M,∈,U〉 has a ∆1-definable global well-order.

Proof:

The (possibly ill-founded) ultrapower N of M by U has a well-order < of M = HNκ+ .

< is represented by the equivalence class [f ].

a < b if {ξ < κ | a f (ξ) b} ∈ U. �

Proposition: The structure 〈M,∈,U〉 has a ∆1-definable truth predicate for 〈M,∈〉.Proof:

Let (κ+)N = OrdM be represented by [f ] in the ultrapower N of M by U.

〈M,∈〉 |= ϕ(a) if {ξ < κ | Hf (ξ) |= ϕ(a)} ∈ U. �

Proposition: The structure 〈M,∈,U〉 has for every n < ω, a Σn-definable truth predicatefor Σn-formulas in the language with U.

To check the truth of a ∆0-formula ϕ(a), we need U ∩ TCl(a).

The truth predicate Tr∆0 (ϕ(x), a) is defined as usual, but with parameterU ∩ TCl(a).

The remaining truth predicates are defined by induction on complexity. �



The structure 〈M,∈,U〉 with some set theory

Suppose κ is inaccessible, M is a simple weak κ-model, with Vκ ∈ M, and U is anM-ultrafilter.

Let ZFC−n denote the theory ZFC with the separation and collections schemes restricted

to Σn-assertions.

Theorem: (G., Schlicht) If 〈M,∈,U〉 |= ZFC−n+1 for n ≥ 1, then for every A ∈ M, there

is a κ-model M ∈ M, with A ∈ M, such that 〈M,∈,U〉 ≺Σn 〈M,∈,U〉 and M ≺ M.

Proof:

Use Σn+1-collection to show that every set X can be extended to a set X closedunder existential witnesses for Σn-formulas in the language with U with parametersfrom X .

Use the well-order < and Σn+1-collection to build unique sequences of length α forα < κ of a chain of models Mξ such that:

I The odd stages ξ + 1 are models closed under existential witnesses for Σn-formulas inthe language with U with parameters from Mξ.

I The even stages ξ + 1 are models elementary in M.

M is correct about κ-models because Vκ ∈ M. �

Proposition: If for every A ∈ M, there is a κ-model M ∈ M, with A ∈ M, such that〈M,∈,U〉 ≺Σn 〈M,∈,U〉, then 〈M,∈,U〉 |= ZFC−

n .



A foray into second-order set theory

Definition

Let ZFC−U denote the theory ZFC− in the language with a unary predicate U.

Let KMU denote the theory Kelley-Morse in the language with a unary predicate Uon classes.

Theorem: (Marek?) The following theories are equiconsistent.

(1) ZFC−U , U is an M-ultrafilter, there is a largest cardinal κ and it is inaccessible.

(2) KMU + U is a normal ultrafilter on Ord.



Baby measurable cardinalsDefinition: (Bovykin, McKenzie, G., Schlicht)

A cardinal κ is weakly n-baby measurable if for every A ⊆ κ, there is a weakκ-model M, with A ∈ M, for which there is a good M-ultrafilter U such that〈M,∈,U〉 |= ZFC−

n .A cardinal κ is n-baby measurable if we replace weak κ-model by κ-model in thedefinition of weakly n-baby measurable cardinal.A cardinal κ is very weakly baby measurable if for every A ⊆ κ, there is a weakκ-model M, with A ∈ M, for which there is a M-ultrafilter U such that〈M,∈,U〉 |= ZFC−.A cardinal κ is weakly baby measurable if for every A ⊆ κ, there is a weak κ-modelM, with A ∈ M, for which there is a good M-ultrafilter U such that〈M,∈,U〉 |= ZFC−.A cardinal κ is baby measurable if we replace weak κ-model by κ-model in thedefinition of weakly baby measurable cardinal.

The n-baby measurable cardinals were introduced by Bovykin and McKenzie.

Theorem: (Bovykin, McKenzie) The following theories are equiconsistent.

(1) ZFC together with the scheme consisting of assertions for every n < ω

“There exist an n-baby measurable cardinal κ such that Vκ ≺Σn V .”

(2) NFUM - A natural strengthening of New Foundations with UrelementsVictoria Gitman Elementary embeddings and smaller large cardinals Oxford 21 / 29


Baby measurable cardinals in the hierarchy

Proposition: A weakly n + 2-baby measurable cardinal is an n-baby measurable limit ofn-baby measurable cardinals.

Proof: Suppose 〈M,∈,U〉 |= ZFC−n+2.

There is a κ-model M ∈ M such that 〈M,∈,U〉 ≺Σn+1 〈M,∈,U〉.〈M,∈,U〉 |= ZFC−

n . �

Theorem: (G., Schlicht) A weakly 0-baby measurable cardinal below which the GCHholds is a limit of 1-iterable cardinals.

Proof: Fix a simple weak κ-model M, with Vκ ∈ M, for which there is a goodM-ultrafilter U such that 〈M,∈,U〉 |= ZFC−

0 .

Use the GCH to show that 2κ = κ+ in the ultrapower N of M by U, and thereforethe well-order < has order-type OrdM .

Use the well-order < and Σ1-collection to show that there are sequences{Mi | i < n} of weak κ-models such that U ∩Mi ∈ Mi+1 for n < ω.

Use Σ1-collection to collect the sequences into a set X .

Build an ill-founded tree inside X of such sequences from X witnessing that U isweakly amenable for some M ∈ M. �



Baby measurable cardinals in the hierarchy (continued)

Theorem: (G., Schlicht) A weakly 1-baby measurable cardinal is a limit of cardinal αthat are α-Ramsey.


1 .

Let N be the ultrapower of M by U.

Suppose [f ] = σ ∈ N is a winning strategy for the challenger in RamseyGκ+

κ (κ).

In M, use U and [f ] to construct a winning run of the game for the judge. �.

Theorem: (G., Schlicht) A weakly 1-baby measurable cardinal below which the GCHholds is strongly Ramsey.


1 .

Use the well-order < and Σ1-collection to show that there are sequences{Mi | ξ < α} of κ-models such that U ∩Mξ ∈ Mξ+1 for α < κ.

Use Σ1-collection to collect the sequences into a set X .

Use Σ1-separation to pick out the sequences from X . �

Theorem: (G., Schlicht) A weakly 2-baby measurable cardinal is strongly Ramsey.



Baby measurable cardinals in the hierarchy

Theorem: (G., Schlicht) A very weakly baby measurable cardinal is n-baby measurablefor every n < ω.

Proof:

Fix a weak κ-model M for which there is an M-ultrafilter U such that〈M,∈,U〉 |= ZFC−.

For every A ∈ M, there is a κ-model M ∈ M such that 〈M,∈,U〉 ≺Σn 〈M,∈,U〉.U ∩ M is a good M-ultrafilter. �

Theorem: (G., Schlicht) A weakly baby measurable cardinal is a limit of very weaklybaby measurable cardinals.

Theorem: (G., Schlicht) A baby measurable cardinal is a limit of weakly babymeasurable cardinals.

Proposition: A measurable cardinal is a limit of baby measurable cardinals.



Games with structures 〈M,∈,U〉

Definition: (G., Schlicht) Suppose α and θ are regular such that ω1 ≤ α ≤ κ and θ > κ.The game weakG θα(κ) is played by the challenger and the judge.At every stage γ < α:

the challenger plays an imperfect κ-model Mγ ≺ Hθ extending his previous moves.

the judge responds with a structure 〈Nγ ,∈,Uγ〉, where Nγ is a κ-model withPMγ (κ) ⊆ Nγ and Uγ is an Nγ-ultrafilter, extending her previous moves.

Let M =⋃γ<αMγ and U =

⋃γ<α Uγ .

The judge wins if she can play for α-many moves such that 〈HMκ+ ,∈,U〉 |= ZFC− and

otherwise the challenger wins.

Note that HMκ+ =

⋃γ<α Nγ .

Definition: (G., Schlicht) The game G θα(κ) is played like weakG θα(κ), but now the judgehas to extend her moves elementarily: if γ < γ, then 〈Nγ ,∈,U〉 ≺ 〈Nγ ,∈,U〉.

Note that HMκ+ =

⋃γ<α Nγ and 〈HM

κ+ ,∈,U〉 |= ZFC−.

Definition: (G., Schlicht) The game strongG θα(κ) is played like weakG θα(κ), but now thejudge has to respond with structures 〈Nγ ,∈,Uγ〉, where Nγ ≺ Hθ is an imperfectκ-model and Uγ is an Nγ-ultrafilter.



Game baby measurable cardinals

Definition: (G., Schlicht)

A cardinal κ is weakly α-game baby measurable for a regular α, with ω1 ≤ α ≤ κ, iffor every A ⊆ κ and arbitrarily large θ there is an imperfect weak κ-model M ≺ Hθ,with A ∈ M, such that M<α ⊆ M for which there is an M-ultrafilter U such that〈HM

κ+ ,∈,U〉 |= ZFC−.

A cardinal κ is α-game baby measurable if we replace the assumption that〈HM

κ+ ,∈,U〉 |= ZFC− with the assumption that for every B ⊆ κ, with B ∈ M, thereis an imperfect κ-model M ∈ M, with B ∈ M, such that 〈M,∈,U〉 ≺ 〈HM

κ+ ,∈,U〉.A cardinal κ is strongly α-game baby measurable if we further strengthen to say thatfor every B ∈ M, there is an imperfect κ-model M ∈ M, with B ∈ M, such that〈M,∈,U〉 ≺ 〈M,∈,U〉.



Games and game baby measurable cardinalsTheorem: (G., Schlicht) The existence of a winning strategy for either player in thegame weakG θα(κ) or the game G θα(κ) is independent of θ.

Theorem: (G., Schlicht) A cardinal κ is weakly α-game baby measurable if and only thechallenger doesn’t have a winning strategy in the game weakG θα(κ) for some/all cardinalsθ. A cardinal κ is α-game baby measurable if and only if the challenger doesn’t have awinning strategy in the game G θα(κ) for some/all cardinals θ.

Theorem: (G., Schlicht) Every weakly β-game baby measurable cardinal is a limit ofcardinals δ > α that are α-game baby measurable for every α < β. An analogous resultholds for α-game measurable cardinals.

Proposition: A weakly ω1-game baby measurable cardinal is a limit of weakly babymeasurable cardinals.

Theorem: (G., Schlicht) A baby measurable cardinal is a limit of cardinals α that areweakly <α-game baby measurable. A weakly κ-game baby measurable cardinal is a limitof baby measurable cardinals.

Theorem: (G., Schlicht) A ω1-game baby measurable cardinal is a limit of cardinals αthat are weakly α-game baby measurable.

Theorem: (G., Schlicht) A measurable cardinal is a limit of cardinals α that are stronglyα-game baby measurable.



Strongly game baby measurable cardinals

Theorem: (G., Schlicht) A cardinal κ is strongly α-game baby measurable if and only ifthe challenger doesn’t have a winning strategy in the game strongG θα(κ) for any θ.

Open Question: Is the existence of winning strategies for either player in the gamestrongG θα independent of θ?

Open Question: Is a strongly β-game baby measurable cardinal a limit of stronglyα-game baby measurable cardinals for α < β?

Open Question: Are strongly α-game baby measurable cardinals stronger than α-gamebaby measurable cardinals?



The hierarchy

weakly compact


ω1-iterable

Ramsey

L

α-Ramsey

strongly Ramsey

super Ramsey

κ-Ramsey

weakly baby measurable

weakly α-game baby measurable

baby measurable

weakly κ-game baby measurable

α-game baby measurable

measurable


victoria gitman · 2020. 10. 10. · victoria gitman elementary embeddings and smaller large...

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