Large cardinals as reflection propertiesof the set-theoretic universe
I. Structural Reflection

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

Three (related) questions

Questions:
1. What is a large cardinal axiom?
2. Why do large cardinal axioms fall into a linear hierarchy, withrespect to (consistency) strength?
3. Is there a single principle that yields in a uniform way thewhole hierarchy of large cardinal axioms?

Three (related) questions

Questions:
1. What is a large cardinal axiom?
2. Why do large cardinal axioms fall into a linear hierarchy, withrespect to (consistency) strength?
3. Is there a single principle that yields in a uniform way thewhole hierarchy of large cardinal axioms?

Three (related) questions

Questions:
1. What is a large cardinal axiom?
2. Why do large cardinal axioms fall into a linear hierarchy, withrespect to (consistency) strength?
3. Is there a single principle that yields in a uniform way thewhole hierarchy of large cardinal axioms?

Infinity as Self-Similarity

Self-similarity• Wikipedia: In mathematics, a self-similar object is exactly orapproximately similar to a part of itself (i.e. the whole has thesame shape as one or more of the parts).
• The Free Dictionary: [Being self-similar means] having asubstructure analogous or identical to an overall structure. In
mathematics, certain geometrical objects such as line
segments and fractals are self-similar to an arbitrary level ofmagnification; many natural phenomena, such as clouds andplants, are self-similar to some degree.

Self-similarity• Wikipedia: In mathematics, a self-similar object is exactly orapproximately similar to a part of itself (i.e. the whole has thesame shape as one or more of the parts).
• The Free Dictionary: [Being self-similar means] having asubstructure analogous or identical to an overall structure. In
mathematics, certain geometrical objects such as line
segments and fractals are self-similar to an arbitrary level ofmagnification; many natural phenomena, such as clouds andplants, are self-similar to some degree.

Dedekind’s definition of infinite set
A set S is said to be infinite when it is similar [i.e.,bijectable] to a proper subset of itself, otherwise it is said
to be finite.1

Richard Dedekind (1831 - 1916)
1

Was sind und was sollen die Zahlen? (1888).

Infinity as self-similarity

Thus, according to Dedekind, what makes a set infinite is not its
size, but its being self-similar. Size is, one might say, a side-effect.
We will argue that the hierarchy of large cardinals, corresponds, inthe context of ZFC, to ever stronger self-similarity properties of the
set-theoretic universe.

Infinity as self-similarity

Thus, according to Dedekind, what makes a set infinite is not its
size, but its being self-similar. Size is, one might say, a side-effect.
We will argue that the hierarchy of large cardinals, corresponds, inthe context of ZFC, to ever stronger self-similarity properties of the
set-theoretic universe.

Infinity as self-similarity

Thus, according to Dedekind, what makes a set infinite is not its
size, but its being self-similar. Size is, one might say, a side-effect.
We will argue that the hierarchy of large cardinals, corresponds, inthe context of ZFC, to ever stronger self-similarity properties of the
set-theoretic universe.

Reflection: V is similar to some (many) of its initial segments
The Reflection Theorem (Levy 1960, Montague 1961):
For every n, there exists a closed and unbounded proper class C (n)of ordinals such that Vα ⪯Σn V , for every α ∈ C (n) .

Can V be self-similar?

V contains many isomorphic copies of itself: for any set a, the classV ⇂ a := {x : x ∩ Vρ(a) = a}is isomorphic to V , via the map ia : V → V ⇂ a defined recursively(with a as a parameter) by:ia (∅) = aia (x) = {ia (y ) : y ∈ x}.
But V cannot be isomorphic to a transitive class different from V ,for any two transitive isomorphic classes must be identical.

Can V be self-similar?

V contains many isomorphic copies of itself: for any set a, the classV ⇂ a := {x : x ∩ Vρ(a) = a}is isomorphic to V , via the map ia : V → V ⇂ a defined recursively(with a as a parameter) by:ia (∅) = aia (x) = {ia (y ) : y ∈ x}.
But V cannot be isomorphic to a transitive class different from V ,for any two transitive isomorphic classes must be identical.

Larger cardinals as critical points of elementary embeddings
A natural weaker requirement: there is an isomorphic copy, V ′ , ofV , different from V and hence not transitive, which is very similar
to a transitive class M, i.e., V ′ is an elementary substructure of M.
Note: if π : V → V ′ is an isomorphism and V ′ is an elementary
substructure of M, then Id ◦π : V → M is an elementary embedding.
• κ is measurable iff it is the critical point of an elementaryembedding j : V → M, with M transitive. (Scott 1961)

Larger cardinals as critical points of elementary embeddings
A natural weaker requirement: there is an isomorphic copy, V ′ , ofV , different from V and hence not transitive, which is very similar
to a transitive class M, i.e., V ′ is an elementary substructure of M.
Note: if π : V → V ′ is an isomorphism and V ′ is an elementary
substructure of M, then Id ◦π : V → M is an elementary embedding.
• κ is measurable iff it is the critical point of an elementaryembedding j : V → M, with M transitive. (Scott 1961)

Larger cardinals as critical points of elementary embeddings
A natural weaker requirement: there is an isomorphic copy, V ′ , ofV , different from V and hence not transitive, which is very similar
to a transitive class M, i.e., V ′ is an elementary substructure of M.
Note: if π : V → V ′ is an isomorphism and V ′ is an elementary
substructure of M, then Id ◦π : V → M is an elementary embedding.
• κ is measurable iff it is the critical point of an elementaryembedding j : V → M, with M transitive. (Scott 1961)

Larger cardinals as critical points of elementary embeddings
A natural weaker requirement: there is an isomorphic copy, V ′ , ofV , different from V and hence not transitive, which is very similar
to a transitive class M, i.e., V ′ is an elementary substructure of M.
Note: if π : V → V ′ is an isomorphism and V ′ is an elementary
substructure of M, then Id ◦π : V → M is an elementary embedding.
• κ is measurable iff it is the critical point of an elementaryembedding j : V → M, with M transitive. (Scott 1961)

More similarity implies larger cardinals

κ is strong if, for every λ > κ there exists an elementary embeddingj : V → M, M transitive, with critical point κ, and such thatVλ ⊆ M. (Dodd-Jensen 1982)
κ is supercompact if, for every λ > κ there exists an elementaryembedding j : V → M, M transitive, with critical point κ, andsuch that j(κ) > λ and M is closed under λ-sequences. (SolovayReinhardt, late 1960’s)

More similarity implies larger cardinals

κ is strong if, for every λ > κ there exists an elementary embeddingj : V → M, M transitive, with critical point κ, and such thatVλ ⊆ M. (Dodd-Jensen 1982)
κ is supercompact if, for every λ > κ there exists an elementaryembedding j : V → M, M transitive, with critical point κ, andsuch that j(κ) > λ and M is closed under λ-sequences. (SolovayReinhardt, late 1960’s)

Local Self-Similarity
The Downwards Löwenheim-Skolem Theorem:
For every infinite structure M in a first-order language L, there
exists N ⪯ M such that |N| ⩽ |L|.

M
N

Note: N need not be of the same kind as M. E.g., if M is transitive,then N need not be so.

Local Self-Similarity
The Downwards Löwenheim-Skolem Theorem:
For every infinite structure M in a first-order language L, there
exists N ⪯ M such that |N| ⩽ |L|.

M
N

Note: N need not be of the same kind as M. E.g., if M is transitive,then N need not be so.

Supercompactness as local second-order self-similarity
Theorem: (Magidor 1971)
The first supercompact cardinal is the first cardinal κ that reflects
every second order formula φ, i.e., given a second order formula φ(in a countable language), for every structure A that satisfies φthere exists a substructure B ⊆ A of cardinality less than κ that
also satisfies φ.

Main question
Question: Can all large cardinal axioms be shown equivalent to some uniform notion of self-similarity?

Kurt Gödel (1906 - 1978)[Quoted in Wang 1996] The universe of sets cannot be uniquely
characterized (i.e., distinguished from all its initial segments) by
any internal structural property of the membership relation init which is expressible in any logic of finite or transfinite type,including infinitary logics of any cardinal number.

Main question
Question: Can all large cardinal axioms be shown equivalent to some uniform notion of self-similarity?

Kurt Gödel (1906 - 1978)[Quoted in Wang 1996] The universe of sets cannot be uniquely
characterized (i.e., distinguished from all its initial segments) by
any internal structural property of the membership relation init which is expressible in any logic of finite or transfinite type,including infinitary logics of any cardinal number.

Structural ReflectionStructural Reflection
A class of relational structures C (of the same kind) is given by someformula φ(x), so thatC = {A = ⟨X , ∈, ⟨Ri ⟩i∈I ⟩ : φ(A)}.

Structural Reflection:
SR κ (C): κ reflects C, i.e., for every A in C there exist B in C ∩ Vκand an elementary embedding from B into A.
AκBStructural Reflection

Theorem:SR κ (Σ1 ) holds for every κ ∈ C (1) (i.e., such that Vκ ⪯Σ1 V ).
Recall: A cardinal κ is supercompact if, for every λ > κ there exists anelementary embedding j : V → M, M transitive, with critical point κ, andsuch that j(κ) > λ and M is closed under λ-sequences.

Theorem: (Magidor 1971, B. 2012)
The following are equivalent:
1. SR(Π1 ) (equivalently, SR(Σ2 ))2. There exists a supercompact cardinal.

Structural Reflection

Theorem:SR κ (Σ1 ) holds for every κ ∈ C (1) (i.e., such that Vκ ⪯Σ1 V ).
Recall: A cardinal κ is supercompact if, for every λ > κ there exists anelementary embedding j : V → M, M transitive, with critical point κ, andsuch that j(κ) > λ and M is closed under λ-sequences.

Theorem: (Magidor 1971, B. 2012)
The following are equivalent:
1. SR(Π1 ) (equivalently, SR(Σ2 ))2. There exists a supercompact cardinal.

Structural Reflection

Theorem:SR κ (Σ1 ) holds for every κ ∈ C (1) (i.e., such that Vκ ⪯Σ1 V ).
Recall: A cardinal κ is supercompact if, for every λ > κ there exists anelementary embedding j : V → M, M transitive, with critical point κ, andsuch that j(κ) > λ and M is closed under λ-sequences.

Theorem: (Magidor 1971, B. 2012)
The following are equivalent:
1. SR(Π1 ) (equivalently, SR(Σ2 ))2. There exists a supercompact cardinal.

Beyond supercompactness

Recall: κ is extendible if for every λ > κ there exists an elementaryembedding j : Vλ → Vµ , some µ, with crit(j) = κ, and j(κ) > λ.

Theorem: (B. 2012)
The following are equivalent:
1. SR(Π2 )2. There exists an extendible cardinal.

Beyond supercompactness

Recall: κ is extendible if for every λ > κ there exists an elementaryembedding j : Vλ → Vµ , some µ, with crit(j) = κ, and j(κ) > λ.

Theorem: (B. 2012)
The following are equivalent:
1. SR(Π2 )2. There exists an extendible cardinal.

For the higher levels (n > 2) we need the notion of C (n) -extendible
cardinal.
Recall: C (n) is the class of cardinals κ such that Vκ ⪯Σn V .

Definition: (B. 2012)κ is C (n) -extendible if for every λ greater than κ there exists anelementary embedding j : Vλ → Vµ , some µ, with crit(j) = κ,j(κ) > λ, and j(κ) ∈ C (n) .
Note: κ is extendible if and only if it is C (1) -extendible.

For the higher levels (n > 2) we need the notion of C (n) -extendible
cardinal.
Recall: C (n) is the class of cardinals κ such that Vκ ⪯Σn V .

Definition: (B. 2012)κ is C (n) -extendible if for every λ greater than κ there exists anelementary embedding j : Vλ → Vµ , some µ, with crit(j) = κ,j(κ) > λ, and j(κ) ∈ C (n) .
Note: κ is extendible if and only if it is C (1) -extendible.

For the higher levels (n > 2) we need the notion of C (n) -extendible
cardinal.
Recall: C (n) is the class of cardinals κ such that Vκ ⪯Σn V .

Definition: (B. 2012)κ is C (n) -extendible if for every λ greater than κ there exists anelementary embedding j : Vλ → Vµ , some µ, with crit(j) = κ,j(κ) > λ, and j(κ) ∈ C (n) .
Note: κ is extendible if and only if it is C (1) -extendible.

For the higher levels (n > 2) we need the notion of C (n) -extendible
cardinal.
Recall: C (n) is the class of cardinals κ such that Vκ ⪯Σn V .

Definition: (B. 2012)κ is C (n) -extendible if for every λ greater than κ there exists anelementary embedding j : Vλ → Vµ , some µ, with crit(j) = κ,j(κ) > λ, and j(κ) ∈ C (n) .
Note: κ is extendible if and only if it is C (1) -extendible.

Theorem: (B. 2012)
The following are equivalent for n ⩾ 1:
1. SR(Πn+1 )2. There exists a C (n) -extendible cardinal.
Corollary:
The following schemata are equivalent:
1. SR
2. There exists a C (n) -extendible cardinal, for every n.
3. There exist a proper class of C (n) -extendible cardinals, forevery n.
4. Vopeňka’s Principle (i.e., for every proper class C of structuresof the same type, there exist A ̸= B in C with an elementaryembedding j : A → B.)

Theorem: (B. 2012)
The following are equivalent for n ⩾ 1:
1. SR(Πn+1 )2. There exists a C (n) -extendible cardinal.
Corollary:
The following schemata are equivalent:
1. SR
2. There exists a C (n) -extendible cardinal, for every n.
3. There exist a proper class of C (n) -extendible cardinals, forevery n.
4. Vopeňka’s Principle (i.e., for every proper class C of structuresof the same type, there exist A ̸= B in C with an elementaryembedding j : A → B.)

SR from Supercompact to Vopěnka’s Principle

ComplexitySRΣ1Π1 , Σ 2
Π2 , Σ 3
Π3 , Σ 4..
.

ZFC
Supercompact
Extendible(2)
C -Extendible..
.

Πn , Σn+1..
.

C (n−1) -Extendible..
.

Πn , all n

VPSR and Vopěnka Cardinals

Definition:
δ is a Vopěnka cardinal if it is inaccessible and for every set C ofstructures of the same type with C ∈ Vδ+1 \ Vδ , there exist
A ̸= B ∈ C with an elementary embedding j : A → B.
Theorem: (B.-Lücke 2023)
The following are equivalent for every uncountable cardinal δ:
1. δ is a Vopěnka cardinal.
2. For every set C of structures of the same type withC ∈ Vδ+1 \ Vδ , there exists a cardinal κ < δ such thatSR κ (C) holds.

SR and Vopěnka Cardinals

Definition:
δ is a Vopěnka cardinal if it is inaccessible and for every set C ofstructures of the same type with C ∈ Vδ+1 \ Vδ , there exist
A ̸= B ∈ C with an elementary embedding j : A → B.
Theorem: (B.-Lücke 2023)
The following are equivalent for every uncountable cardinal δ:
1. δ is a Vopěnka cardinal.
2. For every set C of structures of the same type withC ∈ Vδ+1 \ Vδ , there exists a cardinal κ < δ such thatSR κ (C) holds.

Below supercompactnessSR below a supercompact

Since SR(Σ1 ) is true and SR(Π1 ) is equivalent to the existence ofa supercompact cardinal, if large cardinals smaller than the firstsupercompact are to be characterized as principles of Structural Reflection, then we need to look at either:(1) SR for classes of structures that are Σ1 -definable with additional
Π1 -definable predicates, or(2) Weaker forms of SR.

SR below a supercompact

Since SR(Σ1 ) is true and SR(Π1 ) is equivalent to the existence ofa supercompact cardinal, if large cardinals smaller than the firstsupercompact are to be characterized as principles of Structural Reflection, then we need to look at either:(1) SR for classes of structures that are Σ1 -definable with additional
Π1 -definable predicates, or(2) Weaker forms of SR.

SR(Σ1 (R))

SR(Σ1 (R))

SR(Σ1 (R)) can be as strong as Π1 -SR.
ExampleThe following are equivalent:
1. SR(Σ1 (PwSet))
2. SR(Σ1 (R)), all Π1 predicates R
3. SR(Π1 )4. There exists a supercompact cardinal.

SR(Σ1 (R))

SR(Σ1 (R)) can be as strong as Π1 -SR.
ExampleThe following are equivalent:
1. SR(Σ1 (PwSet))
2. SR(Σ1 (R)), all Π1 predicates R
3. SR(Π1 )4. There exists a supercompact cardinal.

SR(Σ1 (R))

SR(Σ1 (Cd)) implies 0♯ exists, and much more.
The first cardinal witnessing SR(Σ1 (Cd)) may be the first weaklyinaccessible cardinal (Magidor-Väänänen, 2011).
Bounded SR(Σ1 (R)):
SR − (Σ1 (R)) : There exists a cardinal κ that κ-reflects everyΣ1 (R)-definable (with parameters in Hκ ) class C of structures ofthe same type, i.e., for every A ∈ C of cardinality κ there exists
B ∈ C ∩ Hκ and an elementary embedding from B into A.

SR(Σ1 (R))

SR(Σ1 (Cd)) implies 0♯ exists, and much more.
The first cardinal witnessing SR(Σ1 (Cd)) may be the first weaklyinaccessible cardinal (Magidor-Väänänen, 2011).
Bounded SR(Σ1 (R)):
SR − (Σ1 (R)) : There exists a cardinal κ that κ-reflects everyΣ1 (R)-definable (with parameters in Hκ ) class C of structures ofthe same type, i.e., for every A ∈ C of cardinality κ there exists
B ∈ C ∩ Hκ and an elementary embedding from B into A.

SR(Σ1 (R))

SR(Σ1 (Cd)) implies 0♯ exists, and much more.
The first cardinal witnessing SR(Σ1 (Cd)) may be the first weaklyinaccessible cardinal (Magidor-Väänänen, 2011).
Bounded SR(Σ1 (R)):
SR − (Σ1 (R)) : There exists a cardinal κ that κ-reflects everyΣ1 (R)-definable (with parameters in Hκ ) class C of structures ofthe same type, i.e., for every A ∈ C of cardinality κ there exists
B ∈ C ∩ Hκ and an elementary embedding from B into A.

SR below the first measurableLücke (2022) has shown that it is consistent (modulo a large cardinal
consistency-wise weaker than subtleness), that there exists a cardinal< 2ℵ0 witnessing SR − (Σn ), all n < ω. Thus, even SR − cannot
imply the existence of a strongly inaccessible cardinal.
Proposition
The following are equivalent for any class C of structures of thesame type that is closed under isomorphisms:
1. C is Σ1 .
2. There is a Σ1 formula φ(x) of the language of set theory suchthat for every A, A ∈ C if and only if⟨M, ∈⟩ |= φ(B)for some transitive structure ⟨M, ∈⟩ of size |A| that contains
B, where B is isomorphic to A.

SR below the first measurableLücke (2022) has shown that it is consistent (modulo a large cardinal
consistency-wise weaker than subtleness), that there exists a cardinal< 2ℵ0 witnessing SR − (Σn ), all n < ω. Thus, even SR − cannot
imply the existence of a strongly inaccessible cardinal.
Proposition
The following are equivalent for any class C of structures of thesame type that is closed under isomorphisms:
1. C is Σ1 .
2. There is a Σ1 formula φ(x) of the language of set theory suchthat for every A, A ∈ C if and only if⟨M, ∈⟩ |= φ(B)for some transitive structure ⟨M, ∈⟩ of size |A| that contains
B, where B is isomorphic to A.

SR below the first measurable
Definition
A class C of structures of the same type and closed underisomorphisms is Σ1 (R)∗ if there is a Σ1 formula φ(x) of thelanguage LṘ such that for every A, A ∈ C if and only if⟨M, ∈, R ∩ M⟩ |= φ(B)for some transitive structure ⟨M, ∈⟩ of size |A| that contains B,
where B is isomorphic to A.
Bounded SR(Σ1 (R)∗ ):SR − (Σ1 (R)∗ ) : There exists a cardinal κ that κ-reflects everyΣ1 (R)∗ -definable (with parameters in Hκ ) class C of structures ofthe same type, i.e., for every A ∈ C of cardinality κ there exists
B ∈ C ∩ Hκ and an elementary embedding from B into A.

SR below the first measurable
Definition
A class C of structures of the same type and closed underisomorphisms is Σ1 (R)∗ if there is a Σ1 formula φ(x) of thelanguage LṘ such that for every A, A ∈ C if and only if⟨M, ∈, R ∩ M⟩ |= φ(B)for some transitive structure ⟨M, ∈⟩ of size |A| that contains B,
where B is isomorphic to A.
Bounded SR(Σ1 (R)∗ ):SR − (Σ1 (R)∗ ) : There exists a cardinal κ that κ-reflects everyΣ1 (R)∗ -definable (with parameters in Hκ ) class C of structures ofthe same type, i.e., for every A ∈ C of cardinality κ there exists
B ∈ C ∩ Hκ and an elementary embedding from B into A.

Theorem
1. (Lücke) κ is the least weakly inaccessible cardinal iff κ is theleast cardinal witnessing SR − (Σ1 (Cd)∗ ).
2. (Lücke) κ is the least weakly Mahlo cardinal iff κ is the leastcardinal witnessing SR − (Σ1 (Rg )∗ ).
3. κ is the least α-weakly inaccessible cardinal iff κ is the leastcardinal witnessing SR − (Σ1 (Cd, α-WI )∗ ).
4. Suppose κ is such that if γ ⩽ κ is weakly inaccessible, then2δ ⩽ γ for all cardinals δ < γ. Then κ is the least weakly
compact cardinal iff κ is the least cardinal witnessing
SR − (Σ1 (Cd, WC )∗ ).2
2

WC (x, α) is the Π1 relation “α is a limit ordinal and x is a partial orderingwith no chain of order-type α”.

SR from weak inaccessibility to weak compactness

Complexity

SR −SR − +GCH

Σ1 (Cd)∗Σ1 (Cd, α-WI )∗Σ1 (Rg )∗Σ1 (Cd, WC )∗Weakly inaccessible
α-Weakly inaccessible
Weakly Mahlo

Inaccessible
α-Inaccessible
Mahlo
Weakly-compact