Large cardinals as reflection propertiesof the set-theoretic universe
II. Weaker forms of SR
III. Beyond Vopěnka’s Principle

Joan Bagaria
RIMS Workshop on Large Cardinals and the Continuum
24-27 October 2023

Product Structural Reflection
Joint work with Trevor Wilson

Product Structural Reflection
Product Structural Reflection
PSR(C): There exists a cardinal κ such that for every A in C thereexists a set S of structures of the same type as C (although notnecessarily in C) with A ∈ S and an elementary embeddingQQ
j : (C ∩ Vκ ) →S.
Theorem:PSR(Σ1 ) is witnessed by every κ ∈ C (1) .
Theorem: (B.-Wilson 2022)
The following are equivalent:
1. PSR(Π1 )2. There exists a strong cardinal.

Πn -strong cardinals

Definition: (B.-Wilson 2022)
A cardinal κ is λ-Πn -strong if for every Πn -definable (withoutparameters) class A there is an elementary embedding j : V → M,with M transitive, crit(j) = κ, Vλ ⊆ M, and A ∩ Vλ ⊆ j(A).
A cardinal κ is Πn -strong if it is λ-Πn -strong for every ordinal λ.
Theorem: (B.-Wilson 2022)
The following are equivalent for n > 0:
1. PSR(Πn )2. There exists a Πn -strong cardinal.

“Ord is Woodin”
Definition:“ Ord is Woodin” if for every definable A ⊆ V there exists some κwhich is A-strong, i.e., for every γ there is an elementaryembedding j : V → M with crit(j) = κ, γ < j(κ), Vγ ⊆ M, andA ∩ Vγ = j(A) ∩ Vγ .
Theorem: (B.-Wilson 2022)
The following schemata are equivalent:
1. PSR, i.e., PSR(Πn ) for all n.
2. There exists a Πn -strong cardinal, for every n.
3. There exist a proper class of Πn -strong cardinals, for every n.
4. “ Ord is Woodin”.

SR from Strong to Ord is Woodin

ComplexityPSRΣ1Π1 , Σ2Π2 , Σ3
Π3 , Σ4..
.

ZFC
Strong
Π2 -Strong
Π3 -Strong..
.

Πn , Σn+1..
.

Πn -Strong..
.

Πn , all n

Ord is Woodin

PSR and Woodin cardinals

Definition:A cardinal δ is Woodin if it is inaccessible and for every A ⊆ Vδthere exists some κ < δ which is < δ-A-strong, i.e., for every γ < δthere is an elementary embedding j : V → M with crit(j) = κ,γ < j(κ), Vγ ⊆ M, and A ∩ Vγ = j(A) ∩ Vγ .
Theorem: (B.-Lücke 2023)
The following are equivalent for every uncountable cardinal δ:
1. δ is a Woodin cardinal.
2. For every set C of structures of the same type with C ⊆ Vδ ,there exists a cardinal κ < δ such PSR κ (C) holds.

Bounded PSR and measurable cardinals

Bounded Product Structural Reflection
PSR 1 (C): There exists a cardinal κ such that for every A in C ofrank κ + 1 there exists a set S of structures of the same type as C
(although not necessarily in C) with A ∈ S and an elementary
QQembedding j : (C ∩ Vκ ) →S.

Theorem:
The following are equivalent:
1. PSR 1 (Π1 )2. There exists a measurable cardinal.

Below measurability: Weak
Structural Reflection
Joint work with Philip LückeWeak Structural Reflection
Weak Structural Reflection
WSR(C): There exists a cardinal κ such that C ∩ Vκ ̸= ∅, and forevery A in C of cardinality κ, there exists some B in C ofcardinality less than κ and an elementary embedding of B into A.

Definition: (Villaveces 1998)
An inaccessible cardinal κ is strongly unfoldable if for every ordinal
λ and every transitive ZF − -model M of cardinality κ with κ ∈ Mand <κ M ⊆ M, there is a transitive set N with Vλ ⊆ N and anelementary embedding j : M → N with crit(j) = κ and j(κ) ⩾ λ.
First totally indescribable < First strongly unfoldable < First strong

WSR and Strong Unfoldability

Theorem: (B-Lücke 2023)
The following are equivalent for every cardinal κ:
1. κ is either strongly unfoldable or a limit of supercompact
cardinals.
2. κ witnesses WSR(Σ2 ).
Since supercompact cardinals are strongly unfoldable, we get
Corollary:
The following are equivalent for every cardinal κ:
1. κ is the least strongly unfoldable cardinal.
2. κ is the least cardinal that witnesses WSR(Σ2 ).

WSR and Strong Unfoldability
Theorem: (B-Lücke 2023)
The following are equivalent for every cardinal κ:
1. κ is either C (n) -strongly unfoldable or a limit ofC (n−1) -extendible cardinals.
2. κ witnesses WSR(Σn+1 ).
Since C (n) -extendible cardinals are C n+1) -strongly unfoldable, weget
Corollary:
The following are equivalent for every cardinal κ:
1. κ is the least C (n) -strongly unfoldable cardinal.
2. κ is the least cardinal that witnesses WSR(Σn+1 ).

Subtle cardinals
Given a set A of ordinals, a sequence ⟨Eα : α ∈ A⟩ is called an A-list
if Eα ⊆ α holds for every α ∈ A.
Definition: (Jensen-Kunen 1969)
An infinite cardinal δ is subtle if for every δ-list ⟨Eγ : γ < δ⟩ andevery closed unbounded subset C of δ, there exist β < γ in C withEβ = Eγ ∩ β.
First totally indescribable < First subtle < First strongly unfoldable

If δ is subtle, then in Vδ there are stationary-many strongly unfoldable cardinals.

WSR and Subtle Cardinals
Theorem: (B.-Lücke 2023)
The following are equivalent for every uncountable cardinal δ:
1. δ is either subtle or a limit of subtle cardinals.
2. For every set C of structures of the same type withC ∈ Vδ+1 \ Vδ , there exists a cardinal κ < δ that witnesses
WSR(C).
Corollary:
The following are equivalent for every uncountable cardinal δ:
1. δ is the least subtle cardinal.
2. δ is the least cardinal with the property that for every set C ofstructures of the same type with C ∈ Vδ+1 \ Vδ , there exists acardinal κ < δ that witnesses WSR(C).

WSR from Strong to Ord is subtle

Complexity

WSRΣ1Π1 , Σ 2
Π2 , Σ 3
Π3 , Σ 4..
.

ZFC
Strongly unfoldable(2)
C -Strongly unfoldable
C (3) -Strongly unfoldable..
.

Πn , Σn+1..
.

C (n) -Strongly unfoldable..
.

Πn , all n

Ord is subtle

Patterns of Structural Reflection

Vopěnkasupercompact=

Woodinstrong=subtle
strongly unfoldable

Exact SR: Beyond Vopěnka’sPrincipleJoint work with Philip Lücke

Exact Structural Reflection
Given infinite cardinals κ < λ and a class C of structures of the sametype, let
Exact Structural Reflection
ESR C (κ,λ): For every A ∈ C of rank λ, there exists some B ∈ C ofrank κ and an elementary embedding form B into A.
We let Γ (P)-ESR(κ, λ) denote the statement that ESR C (κ, λ) holds
for every class C of structures of the same type that is Γ -definablewith parameters from P.
Σ0 -ESR(κ, λ) holds for some κ < λ ⇒ a♯ exists for every real a.

Exact Structural Reflection
The general ESR principle restricted to classes of structures that areclosed under isomorphic images is just equivalent to V P:
TheoremThe following schemata of sentences are equivalent:
1. For every class C of structures of the same type that is closed
under isomorphic images, there is a cardinal κ with theproperty that ESR C (κ, λ) holds for all λ > κ.
2. V P.
However, even the principle Π1 -ESR(κ, λ) holding for some κ < λalready implies the existence of large cardinals, the weakly exact
cardinals, whose consistency strength is beyond that of V P.

Weakly exact cardinals

Definition
Given n > 0, an infinite cardinal κ is weakly n-exact for acardinal λ > κ if for every A ∈ Vλ+1 , there exists a transitive,Πn (Vκ+1 )-correct set M with Vκ ∪ {κ} ⊆ M, a cardinal
λ ′ ∈ C (n−1) greater than ℶλ and an elementary embeddingj : M → Hλ ′ with j(κ) = λ and A ∈ ran(j).
If we further require that j(critj) = κ, then we say that κ is weakly
parametrically n-exact for λ.

Weakly exact cardinals
TheoremThe following statements are equivalent for all κ and all n > 0:
1. κ is the least cardinal such that Πn (Vκ )-ESR(κ, λ) holds forsome λ.
2. κ is the least cardinal that is weakly n-exact for some λ.
3. κ is the least cardinal that is weakly parametrically n-exact forsome λ.
The ESR principles for Πn -definable and Σn+1 -definable classes ofstructures are not equivalent. For Σn -definable classes, the relevant
large cardinals are the exact cardinals:

Exact cardinals

Definition
Given a natural number n, an infinite cardinal κ is n-exact forsome cardinal λ > κ if for every A ∈ Vλ+1 , there exists a cardinalκ ′ ∈ C (n) greater than ℶκ , a cardinal λ ′ ∈ C (n+1) greater than λ,an X ⪯ Hκ ′ with Vκ ∪ {κ} ⊆ X , and an elementary embeddingj : X → Hλ ′ with j(κ) = λ and A ∈ ran(j).
If we further require that j(crit(j)) = κ holds, then we say that κ isparametrically n-exact for λ.

Exact cardinals

Theorem
The following are equivalent for all cardinals κ and all n > 0:
1. κ is the least cardinal such that Σn+1 (Vκ )-ESR(κ, λ) holds forsome λ.
2. κ is the least cardinal that is n-exact for some λ.
3. κ is the least cardinal that is parametrically n-exact for someλ.

The strength of ESR

The strength of weakly n-exact and n-exact cardinals, and theircorresponding equivalent forms of ESR, goes beyond V P, for they
imply the existence of almost huge cardinals.
Recall that a cardinal κ is almost huge if there exists a transitiveclass M and a non-trivial elementary embedding j : V → M with
critj = κ and <j(κ) M ⊆ M.
We then say that a cardinal κ is almost huge with target λ if thereexists such a j with j(κ) = λ.

The strength of ESR• If κ is either parametrically 0-exact for some λ, or weakly
parametrically 1-exact for some λ, then the set of cardinals
that are almost huge with target κ is stationary in κ.
• If κ is parametrically 0-exact for some λ, then it is almost
huge with target λ.

The strength of ESR• If κ is huge1 with target λ, then it is weakly parametrically
1-exact for λ. Hence, Π1 (Vµ )-ESR(µ, ν) holds for someµ ⩽ κ and ν > µ.
Moreover, Π1 (Vκ )-ESR(κ, λ ′ ) holds in Vλ , for some λ ′ .
• However, if κ is the least huge cardinal, then κ is not 1-exact
for any cardinal λ > κ.

1

κ is huge if there exists a transitive class M and a non-trivial elementaryembedding j : V → M with critj = κ and j(κ) M ⊆ M.

Consistency upper bounds

The best upper bound so far for the consistency strength of weakly
n-exact cardinals κ, for some λ, is given by the following:
Proposition
If κ is an almost 2-huge cardinal,2 then there exist aninaccessible cardinal λ > κ and ρ > λ such that Vρ is a model of
ZFC and, in Vρ , the cardinal κ is weakly parametrically n-exact for
λ, for all n > 0. Hence, in Vρ , ESR C (κ, λ) holds for all C that are
definable with parameters in Vκ .
2I.e., there is an elementary embedding j : V → M with M transitive,
2crit(j) = κ, and <j (κ) M ⊆ M.

Consistency upper bounds

The best upper bound so far for the existence of n-exact cardinals
cardinals κ, for a proper class of λ, is given by the following:
Proposition
Let κ be an I3-cardinal3 If i, m, n < ω, then in Vδ the cardinalj i (κ) is parametrically n-exact for j i+m+1 (κ). Hence, in Vδ ,ESR C (κ, λ) holds for all C definable with parameters in Vκ , for aproper class of λ.
3I.e., the critical point of a non-trivial elementary embedding j : Vδ → Vδ , forsome limit ordinal δ. Then Vδ |= ZFC and the sequence ⟨j m (κ) | m < ω⟩ iscofinal in δ.

ESR and (weakly) exact cardinals

ComplexityESRΠ1

κ weakly (param.) 1-exact for some λ > κ(Between almost huge and huge)κ weakly (param.) n-exact for some λ > κ(Consistency-wise below 2-huge, all n)κ (param.) n-exact for some λ > κ(Consistency-wise below 2-huge, all n)

Πn
Σn+1

References

Bagaria, J., Wilson, T. M., The Weak Vopenka Principle for definable classes of structures. The Journal of Symbolic Logic, 88(1),145-168 (2022).
Bagaria, J., Lücke, P., Huge Reflection. Annals of Pure and AppliedLogic. Volume 174, Issue 1, January 2023, 103-171.
Bagaria, J., Large Cardinals as Principles of Structural Reflection.
Bulletin of Symbolic Logic, 29(1), 19-70 (2023).
Bagaria, J., Lücke, P., Patterns of Structural Reflection in the Large
Cardinal Hierarchy. Submitted. https://arxiv.org/abs/2210.07120