Ground model definability in ZF

Ground model definability in

ZF

Victoria Gitman

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

JMM 2020

Choiceless set theory and related areas January 15, 2019

Ground model definability

More than 3 decades after the introduction of forcing, the following fundamental question was asked and answered.

Question: Is the universeV definable in its forcing extensionsV[G]?

Theorem: (Laver (2007), Woodin (2004)) A universeV isuniformly definablein its

forcing extensions1_{with aparameter fromV.}

There is a single formulaϕ(x,y)such that wheneverV[G] is a forcing extension ofV,

there isa∈V such that

V ={x∈V[G]|V[G]|=ϕ(x,a)}.

Cover and approximation properties

Definition(Hamkins, 2003) Suppose thatV ⊆W are transitive models of (some

fragment of)ZFCandδ is a cardinal inW.

The pairV ⊆W satisfies theδ-cover propertyif for everyA∈W withA⊆V and

|A|W

< δ, there isB∈V withA⊆B and|B|V

< δ.

“Every set of size less thanδinW that is contained inV can be covered by a set of size less thanδinV.”

The pairV ⊆W satisfies theδ-approximation propertyif wheneverA∈W with

A⊆V andA∩a∈V for everyaof size less thanδinV, thenA∈V.

“If a set inW is contained inV andV has all pieces of it of size less thanδ, thenV has the set.”

Theorem: (Hamkins) SupposeM,M0, andN are transitive models of (a fragment of)

ZFCsuch that:

δ is regular inN,

the pairsM⊆NandM0⊆N have theδ-cover andδ-approximation properties,

PM(δ) =PM0(δ),

(δ+)M = (δ+)N.

ThenM=M0.

The formula defining

V

SupposeV[G]is a forcing extension by a posetP.

Let|P|=γ. Letδ=γ+_.

LetZ∗be a certainfinite fragment ofZFC.

Say that a transitive modelM∈V[G]isgoodif

I M|=Z∗,

I (δ+₎M _{= (δ}+₎V[G]_,

I PM(δ) =PV(δ).

Lemma

The pairVγ⊆V[G]γ has theδ-cover andδ-approximation propertieswhenever they

satisfyZ∗.

There areunboundedlymany ordinalsγsuch thatVγ andV[G]γsatisfyZ∗.

ϕ(x,PV(δ)) :=

x is an element of agood modelMof heightγδsuch thatV[G]γ|=Z∗and the pair

Set Theoretic Geology

Ground model definability makes it possible to study thestructure of all ground models

of aZFC-universe in afirst-order context.

Theorem: (Reitz, 2006) There is a formulaψ(x,y)such that for every setr, the classWr ={x |ψ(x,r)}is agroundofV,

for everygroundW ofV, there is a setr such thatW ={x |ψ(x,r)}.

Several new and interesting inner models were introduced and studied using the definability of grounds.

Definition: Themantleis theintersection of all groundsofV.

Theorem: (Usuba, 2017) Themantleis amodel ofZFC.

Theorem: (Usuba, 2018) If there exists anextendible cardinal, then themantleis a

(smallest)groundofV.

Ground model definability in

ZF

Cover and approximation properties arguments do not easily generalize to choiceless models because they are fundamentally tied up with existence of cardinalities.

Open Question: Does (uniform) ground model definability hold forZF-universes? Partial results have used (combinations of) two incremental approaches.

Isolate a class of partial orders for which ground model definability holds.

Ground model definability in

ZF

+

DC

Definition: (L´evy, 1964) The choice principleDCδasserts that for any non-empty setS

and anybinary relationR, if for each sequences∈S<δ _{there is ay ∈Ssuch thats R y,}

then there is a functionf :δ→S such thatf αR f(α)for allα < δ.

“We can makeδ-many dependent choices along any relation without terminal nodes.” Definition: A posetPadmits a gap at a cardinalδ if it has the formR∗Q˙ where

|R|< δ(in particular,Ris well-orderable) is a non-trivial forcing,

R“ ˙Qis≤δ-strategically closed”.

Posets admitting a gap atδ preserveDCδ to the forcing extension.

Theorem: (G., Johnstone, 2014) SupposeV |=ZF+DCδandPis a posetadmitting a

gap atδ. Then every forcing extensionV[G]byPsatisfiesDCδ.

Theorem: (G., Johnstone, 2014) A universeV |=ZF+DCδ isuniformly definablein all

its forcing extensions byposets admitting a gap atδ.

There is a single formulaϕ(x,y)such that wheneverV |=ZF+DCδ andV[G] is a

forcing extension ofV by a posetPadmitting a gap atδ, then

V ={x ∈V[G]|V[G]|=ϕ(x,PV(δ))}.

L¨

owenheim-Skolem cardinals

Definition: An uncountable cardinalκis aLowenheim-Skolem cardinal(LS) if for every

γ < κ≤αandx∈Vα, there isβ > αandX≺Vβ such that

Vγ⊆X,

x ∈X,

thetransitive collapse ofX belongs toVκ,

(X∩Vα)Vγ ⊆X.

Proposition: SupposeV |=ZFC. Then theLScardinalsare precisely thei-fixed point

cardinals.

Definition: (Woodin) SupposeV |=ZF. A cardinalκissupercompactif for every

α≥κ, there isβ≥α, a transitive setNand an elementary embeddingj:Vβ →N such

thatcrit(j) =κ,α <j(κ), andNVα_⊆N.

Proposition: SupposeV |=ZF. Then everysupercompact cardinalis anLScardinal.

Theorem: (Usuba, 2019) The assertion that there is aproper class ofLScardinalsis

absolutebetweenV and its forcing extensions.

Corollary: IfACis forceable overV |=ZF, thenV has a proper class ofLS-cardinals.

Universes without

LS

cardinals

Theorem: (Usuba, 2019) SupposeV |=ZF. Ifκis anLScardinal, then theclub filter onκ+ isκ-complete.

Theorem: (Karagila, 2018) Every universeV |=ZFC+GCHhas anextension to a universeW |=ZFwith thesame cofinalitiesin which theclub filter is notσ-complete for any regular cardinalδ.

Corollary: It is consistent that there are universesV |=ZFwithout anyLScardinals.

Ground model definability in

ZF

+ proper class of

LS

cardinals

Theorem: (Usuba, 2019) A universeV |=ZF+ proper class ofLScardinalsisuniformly definablein all its forcing extensions.

There is a single formulaϕ(x,y)such that whenever

V |=ZF+ proper class ofLScardinals

andV[G]is a forcing extension by a posetP∈Vκfor anLScardinalκ, then

V ={x∈V[G]|V[G]|=ϕ(x,Vκ)}.

The proof is a modification of the cover and approximation properties arguments, where cardinality is replaced by a rough measure on sets:

(Usuba) Define for a setx that themeasure||x||is the least ordinalαsuch that

there is a surjection fromVα ontox.

Corollary: Uniform ground model definability holds for the following universes.

Universes satisfyingZF+ proper class of supercompact cardinals.