Perfect Set Theorems at singular cardinals

Philipp Moritz Lücke
Universität Hamburg

RIMS Set Theory Workshop 2023, 27. October 2023

Joint work in progress withVincenzo Dimonte (Udine) andMartina Iannella (Vienna).

Remember that a set of real numbers has the Perfect Set Property(PSP) if it is either countable or contains a non-empty closed subset
without isolated points.
Cantor showed that every closed set of reals has the PSP and Suslin
extend this implication to all analytic sets.
While Gödel showed that the existence of a co-analytic set of realswithout the PSP is consistent, seminal results of Shelah–Woodin andMartin–Steel showed that the existence of large cardinals implies thatevery set of reals in L(R) has the PSP.

There has recently been an interest to develop a generalized descriptive set
theory that allows the study of definable objects of higher cardinalities.
While it is known that several key results of the classical theory cannot bedirectly generalized to all higher cardinalities, several settings with rich
structure theories for definable sets were isolated.
The work presented in this talk contributes to the study of one of thesesettings that originates in Hugh Woodin‘s work on large cardinalassumptions close to the Kunen Inconsistency.

Definition
A non-trivial elementary embedding j : L(Vλ+1 ) −→ L(Vλ+1 ) forsome ordinal λ is an I0-embedding if crit(j) < λ holds.

Given a non-empty set X and an infinite cardinal µ, we equip the set µ X of
all functions from µ to X with the topology whose basic open sets consistsof all functions that extend a given function s : ξ −→ X with ξ < µ.
In addition, we equip the set P(ν) of all subsets of an infinite cardinal νwith the topology whose basic open sets consist of all subsets of ν whoseintersection with a given ordinal η < ν is equal to a fixed subset of η.
Finally, we say that a map ι : X −→ Y between topological spaces is aperfect embedding if it induces a homeomorphism between X and thesubspace ran(ι) of Y .

Theorem (Cramer, Shi & Woodin)
If j : L(Vλ+1 ) −→ L(Vλ+1 ) is an I0-embedding and X is a subsetof P(λ) of cardinality greater than λ that is an element of L(Vλ+1 ),then there is a perfect embedding ι : ω λ −→ P(λ) with ran(ι) ⊆ X.

Question
Do weaker large cardinal assumptions suffice to derive the above conclusion for smaller classes of definable subsets of P(λ)?

Definition
A class X is definable by a formula ϕ(v0 , . . . , vn ) and parametersz0 , . . . , zn−1 if
X = {y | ϕ(y, z0 , . . . , zn−1 )}.

Definition• A formula in the language of set theory is a Σ0 -formula if it iscontained in the smallest collection of L∈ -formulas that contains
all atomic formulas and is closed under negation, disjunction andbounded quantification.
• Given n < ω, an L∈ -formula is a Σn+1 -formula if it is of theform ∃x ¬ϕ(x) for some Σn -formula ϕ.

Theorem (L.–Müller)
If λ is a limit of measurable cardinals and X is a subset of P(λ)
of cardinality greater than λ that is definable by a Σ1 -formulawith parameters in Vλ ∪ {λ}, then there is a perfect embeddingι : cof(λ) λ −→ P(λ) with ran(ι) ⊆ X.

Theorem (L.–Müller)
Let λ be a singular strong limit cardinal with the property thatfor every subset of P(λ) of cardinality greater than λ that isdefinable by a Σ1 -formula with parameters in Hκ ∪ {κ} contains
the range of a perfect embedding of cof(λ) λ into P(λ). Then
there is an inner model with a sequence of measurable cardinalsof length cof(λ).

Note that, if λ is an infinite cardinal, then L(Vλ+1 ) contains all
subsets of P(λ) that are definable by a Σ1 -formula with parameters in
Hλ+ , because Hλ+ is contained in L(Vλ+1 ) and all Σ1 -formulas areabsolute between V and Hλ+ .
Question
Can we derive a stronger Perfect Set Theorem at limits of measurable cardinals?
What happens if we allow other simple parameters, like Vλ or acofinal ω-sequence in λ, in our Σ1 -definitions?

Theorem (Dimonte–Iannella–L.)
If ~λ is a strictly increasing sequence of measurable cardinals with limit
λ, then the following statements hold in an inner model M containing~λ:• The sequence ~λ consists of measurable cardinals.
• If ~ν is a strictly increasing ω-sequence of regular cardinals withlimit λ, then there is a subset of P(λ) that does not contain therange of a perfect embedding of ω λ into P(λ) and is definable
by a Σ1 -formula with parameters in Hℵ1 ∪ {~ν }.

Definition
An elementary embedding j : V −→ M with critical sequence hλn | n < ωi is an I2-embedding if Vλ ⊆ M , where
λ = supn<ω λn .

Theorem (Dimonte–Iannella–L.)
Let j : V −→ M be an I2-embedding with critical sequence~λ = hλn | n < ωi and set λ = supn<ω λn . If X is a subsetof P(λ) of cardinality greater than λ that is definable by a Σ1 formula with parameters in Vλ ∪ {Vλ , ~λ}, then there is a perfectembedding ι : ω λ −→ P(λ) with ran(ι) ⊆ X.

Theorem (Dimonte–Iannella–L.)
If j : V −→ M is an I2-embedding with critical sequence ~λ =hλn | n < ωi and λ = supn<ω λn , then the following statements
hold in an inner model:• There is an I2-embedding whose critical sequence has supremum λ.
• There is a subset of P(λ) that does not contain the range
of a perfect embedding of ω λ into P(λ) and is definable bya Σ1 -formula with parameters in P(λ).

The results discussed above suggest the possibility of studying large cardinalassumptions inducing singular cardinals λ of countable cofinality throughthe provable regularity properties of simply definable subsets of P(λ).
More specifically, they suggest we can assign definable subsets of Vλ+1 tothese axioms in a way that ZFC proves that all subsets of P(λ) ofcardinality greater than λ that are definable by Σ1 -formulas usingparameters from the given subset have the Perfect Set Property.
Note that, since this approach is based on provable implications and notconsistency strength, it is less affected by the current technical limitationsof inner model theory and therefore provides a new angle to study strong
large cardinal axioms.

Descriptive properties ofI2-embeddings

In the following, we outline the proof of our main result:
Theorem (Dimonte–Iannella–L.)
Let j : V −→ M be an I2-embedding with critical sequence~λ = hλn | n < ωi and set λ = supn<ω λn .
If X is a subset of P(λ) of cardinality greater than λ that isdefinable by a Σ1 -formula with parameters in Vλ ∪ {Vλ , ~λ}, thenthere is a perfect embedding ι : ω λ −→ P(λ) with ran(ι) ⊆ X.
We will in fact show that the above conclusion holds for a larger classof parameters that we will now define.

Classical results of Martin show that I2-embeddings j : V −→ M are ωiterable, i.e. there exists a commuting systemhhMαj | α ≤ ωi, hj : Mαj −→ Mβj | α ≤ β ≤ ωii
of inner models and elementary embeddings with:• M0j = V and j0,1 = j.
S• If n < ω, then jn+1,n+2 = {jn,n+1 (jn,n+1  Vα ) | α ∈ Ord}.
• hMωj , hjn,ω | n < ωii is a direct limit ofjhhMnj | n < ωi, hjm,n : Mm−→ Mnj | m ≤ n < ωii.

If ~λ = hλn | n < ωi is the critical sequence of j and λ = supn<ω λn , then:j• Given m ≤ n < ω, we then have Vλ ⊆ Mωj ⊆ Mnj ⊆ Mm
,crit(jn,n+1 ) = λn = jm,n (λm ), jm,n (λ) = λ and jn,ω (λn ) = λ.
j

• j0,ω (λ+ ) = λ+ and (2λ )Mω < λ+ .
j ~• ~λ is Prikry-generic over Mωj and hence (2λ )Mω [λ] < λ+ .

Theorem (Laver)
Let j : V −→ M be an I2-embedding with critical sequencehλn | n < ωi and set λ = supn<ω λ.
If d ∈ Vλ and r : d −→ Ord is a function, then the function
j0,ω ◦ r : d −→ Ord
is an element of Mωj .

Using Laver’s result, we will be able to prove a strengthening of the above
Perfect Set Theorem.

Theorem (Dimonte–Iannella–L.)
Let j : V −→ M be an I2-embedding with critical sequence~λ = hλn | n < ωi and let N be an inner model of ZFC withMωj ∪ {~λ} ⊆ N . Set λ = supn<ω λn .
If X is a subset of ω λ with |X| > (2λ )N that is definable byNa Σ1 -formula with parameters in Vλ+1
, then there is a perfectωembedding ι : λ −→ C(~λ) with ran(ι) ⊆ X.

N
• A subset of ω λ is definable by a Σ1 -formula with parameters in Vλ+1
N .
iff it is definable over Vλ by a Σ12 -formula with parameters in Vλ+1

• A subset of ω λ × ω λ that is definable over Vλ by a Σ11 -formula withN can be represented as the projection p[T ] of theparameters in Vλ+1
set [T ] of all cofinal branches through a subtree T ∈ N of (<ω Vλ )3 .
• We can build a Shoenfield tree for the Σ12 -subset of ω λ defined by T .
Let STV denote the Shoenfield tree of T in V and let STN denote theShoenfield tree of T in N .
• Then STN ⊆ STV and we can use Laver’s theorem to find an embeddingof STV into STN that is the identity on the first coordinate.
• We then know that p[STN ]V = p[STV ]V .

LemmaLet ~λ = hλn | n < ωi be a strictly increasing sequence of infinite
cardinals with limit λ and let T ⊆ <ω a × <ω b be a tree such that p[T ]does not contain the range of a perfect embedding of ω λ into ω a.
If N is an inner model with Vλ ∪ {T, ~λ} ⊆ N , then p[T ]V ⊆ N .

• Assume that p[STV ]V has cardinality greater than (2λ )N .
• Then p[STN ]V = p[STV ]V * N .
• The lemma shows that p[STV ]V contains the range of a perfectembedding of ω λ into itself.

Open questions

Question
Assume that j : Vλ+1 −→ Vλ+1 is an I1-embedding and X is a subsetof P(λ) of cardinality greater than λ that is definable in Hλ+ .
Does X contain the range of a perfect embedding of ω λ into P(λ)?

QuestionLet ~λ = hλn | n < ωi be a strictly increasing sequence of cardinals
with limit λ such that each λn is <λ-supercompact.
If X is a subset of P(λ) of cardinality greater than λ that is definable
by a Σ1 -formula with parameters in Vλ ∪ {~λ}, does X contain therange of a perfect embedding of ω λ into P(λ)?

Thank you for listening!