Large cardinal

A large cardinal property strengthens uncountability by imposing combinatorial, logical, measure-like, or elementary-embedding structure whose existence is not provable in ZFC if ZFC is consistent.

κ

is supercompact if for every cardinal

λ \geq κ

there is an elementary embedding

j : V \to M

into a transitive class with critical point of an elementary embedding

κ

j (κ) > λ

, and

M^{λ} \subseteq M

. This closure supplies every sufficiently small rank segment needed to transfer existential witnesses into

M

Supercompact Sigma-two downward reflection ( $V ⊨ σ ⟹ V_{κ} ⊨ σ (σ \in Σ_{2})$ )

 0  0

κ

is a supercompact cardinal, each true

Σ_{2}

statement, allowing parameters in

V_{κ}

, holds in

V_{κ}

. Write it as an existential witness followed by a

Π_{1}

assertion. Choose a supercompact elementary embedding whose transitive target contains the witness in

V_{j (κ)}

. Downward absoluteness of the universal bounded-matrix assertion puts the witness in that rank segment, and elementarity reflects this to

V_{κ}

Mahlo cardinal ( $κ Mahlo$ )

 0  0

An inaccessible cardinal

κ

whose inaccessible ordinals below it form a stationary set. Equivalently, every predicate expansion of

V_{κ}

has an inaccessible elementary rank level below

κ

Mahloness is downward absolute to the constructible universe

 0  0

κ

is Mahlo in

V

, then it is Mahlo in

L

. Inaccessibility is downward absolute, and every club in

L

remains a club in

V

, where it meets a

V

-inaccessible ordinal.

Weakly inaccessible cardinal ( $cf (κ) = κ and κ is an uncountable limit cardinal$ )

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A weakly inaccessible cardinal is an uncountable regular cardinal that is a limit cardinal. It need not be a strong limit cardinal. Under the Generalized continuum hypothesis, it is a strongly inaccessible cardinal, since every smaller infinite

λ

satisfies

2^{λ} = λ^{+} < κ

Model of ZFC without weakly inaccessible cardinals

 0  0

Consistency of ZFC implies consistency of ZFC with the Generalized continuum hypothesis and no weakly inaccessible cardinals. Pass to the constructible universe. If it has an inaccessible, cut at its least one; that rank segment still models ZFC and the Generalized continuum hypothesis, but has no inaccessibles. In this model the singular cardinal enumeration is continuous at every nonzero limit ordinal index. This is a relative-consistency construction, not a deduction of a transitive model from bare consistency.

Strong limit cardinal

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A cardinal

κ

is a strong limit when

2^{λ} < κ

for every cardinal

λ < κ

Singular strong limit power-set identity ( $2^{λ} = λ^{cf (λ)}$ )

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For a singular strong limit cardinal

λ

, let

μ = cf (λ)

and choose cofinal cardinals

λ_{i} < λ

. Subset restrictions inject

P (λ)

into

\prod_{i < μ} P (λ_{i})

, whose cardinal is at most

λ^{μ}

by the strong-limit property. Conversely

λ^{μ} \leq (2^{λ})^{μ} = 2^{λ}

. This equality alone does not imply the singular cardinals hypothesis.

Strongly inaccessible cardinal

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A strongly inaccessible cardinal is an uncountable regular strong limit cardinal.

Inaccessible limit of inaccessibles

 0  0

An inaccessible cardinal

ι

is an inaccessible limit of inaccessibles if the inaccessible cardinals below it are unbounded in

ι

. A rank cutoff at the first inaccessible limit of inaccessibles shows that existence of unboundedly many inaccessibles does not prove existence of such a limit, provided the former theory is consistent.

Rank cutoff at the first inaccessible limit of inaccessibles ( $V_{ι} ⊨ ZFC + \infty IC + \neg ILI$ )

 0  0

ι

is the least inaccessible limit of inaccessibles,

V_{ι}

models ZFC and internally has unboundedly many inaccessible cardinals, while none of its ordinals is an inaccessible limit of inaccessibles. Rank agreement makes inaccessibility below

ι

absolute. If a model of unboundedly many inaccessibles has no such limit in the first place, no cutoff is needed. This proves the corresponding relative-consistency nonimplication.

Keisler extension property

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An inaccessible cardinal

κ

has the Keisler extension property when there is a proper transitive set

X ⊋ V_{κ}

for which

(V_{κ}, \in) ≺ (X, \in) .

(1)

Such a

κ

is not the least inaccessible cardinal:

X

regards

κ

as inaccessible, elementarity reflects the existence of an inaccessible into

V_{κ}

, and strong-inaccessibility absoluteness from rank agreement makes the resulting witness genuinely inaccessible below

κ

Worldly cardinal

 1  0

A cardinal

κ

is worldly when

V_{κ} ⊨ ZFC

. Every strongly inaccessible cardinal is worldly, but worldly cardinals can be singular.

Worldly cardinals below an inaccessible cardinal

 0  0

Below any uncountable strongly inaccessible cardinal

κ

, the worldly cardinals contain a closed unbounded subset of

κ

and hence have cardinality

κ

. The structure

V_{κ}

models ZFC. Reflect each finite group from an enumeration of all first-order formulas to a closed unbounded subset of

κ

, using

∣ V_{α} ∣ < κ

and regularity for the witness bounds. The countable intersection remains closed unbounded, so its ranks are elementary substructures of

V_{κ}

. Intersect with the closed unbounded subset of infinite cardinal numbers below

κ

. Uncountability is essential: the regular strong limit

ω

has no worldly cardinal below it.

Nested transitive models from a worldly cardinal

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A worldly cardinal permits a countable transitive model

M

of ZFC and a transitive model

N

of ZFC with

M \in N

∣ N ∣ = ℵ_{1}

, and

(ω_{1})^{N} = ω_{1}

. First collapse a countable elementary substructure of

V_{κ}

. Then collapse an

ℵ_{1}

-sized Skolem hull containing every countable ordinal and every member of

M

, as well as

M

itself. The transitive collapse fixes transitive subsets lemma preserves

M

and those ordinals.

Arithmetic absoluteness for a rank-initial model

 0  0

V_{κ} ⊨ ZFC

, it has the standard natural numbers and standard finite proofs. Every arithmetical sentence, including a formal consistency statement, therefore has the same truth value in

V_{κ}

and the universe.

Kappa-satisfiable theory ( $κ$ -satisfiable)

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A theory is

κ

-satisfiable when each of its subtheories of cardinality less than

κ

has a model.

Least-occurrence order on cardinal properties ( $<_{1}$ )

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For cardinal properties

Φ

and

Ψ

that occur, let

ι_{Φ}

and

ι_{Ψ}

be their least witnesses. Define

Φ <_{1} Ψ

when

ZFC + Φ C + Ψ C ⊢ ι_{Φ} < ι_{Ψ} .

(1)

This relation need not be transitive, because the two implications can be proved under different joint existence assumptions.

Nontransitivity of the least-occurrence order on cardinal properties

 0  0

Let

I, W, M

mean inaccessible, weakly compact, and measurable, and define

Θ (κ) ⟺ (W C \land M (κ)) \lor (\neg W C \land I (κ)) .

(1)

Then

I <_{1} W

and

W <_{1} Θ

. Subject to the consistency of an inaccessible without a weakly compact cardinal, there is also a model in which

Θ = I

, so

I \neq <_{1} Θ

Omega-measurable cardinal ( $ω -measurable$ )

 0  0

A cardinal is omega-measurable if it carries a countably complete ultrafilter that is a nonprincipal ultrafilter. Countable completeness rules out countable underlying sets. Such a measure is weaker than the cardinal's own full completeness requirement, but the least omega-measurable cardinal is measurable.

Least omega-measurable cardinal is measurable

 0  0

κ

is the least omega-measurable cardinal and

U

witnesses it, failure of

κ

-completeness partitions a member of

U

into fewer than

κ

fibres, none belonging to

U

. The pushforward ultrafilter under the fibre-index map is then a nonprincipal countably complete measure on a smaller cardinal, a contradiction. Thus

U

κ

-complete and

κ

is a measurable cardinal.

Measurable cardinal is a strong limit cardinal

 0  0

Let

U

be a nonprincipal

κ

-complete ultrafilter on

κ

. If

λ < κ

and

(A_{α})_{α < κ}

were distinct subsets of

λ

, then for every

ξ < λ

choose the

U

-large side of the partition according to whether

ξ \in A_{α}

. Their intersection is

U

-large by

κ

-completeness, but all its indices label the same subset of

λ

, so it has at most one member, contradicting nonprincipality. Thus

2^{λ} < κ

Measurable cardinal is one-strong

 0  0

The ultrapower embedding

j : V_{λ} \to M

associated with a measure on

κ < λ

has critical point

κ

. For every

A \in V_{κ + 1}

A = j (A) \cap V_{κ},

(1)

V_{κ + 1} \subseteq M

. Hence every measurable cardinal is 1-strong.

Two measurable cardinals under an ultrapower embedding

 0  0

Let

κ_{0} < κ_{1} < λ

, where both

κ_{i}

are measurable and

λ

is inaccessible, and let

j_{i} : V_{λ} \to M_{i}

be ultrapower embeddings by measures on

κ_{i}

. Then

j_{0} (κ_{1}) = κ_{1}, j_{1} (κ_{0}) = κ_{0} .

(1)

The second equality follows from the critical point. For the first, regularity gives continuity of

j_{0}

κ_{1}

, while strong-limitness bounds

j_{0} (β) < κ_{1}

for every

β < κ_{1}

Moved critical point is not an ambient cardinal under GCH

 0  0

For a measure ultrapower

j : V_{λ} \to M

with critical point

κ

, there are at most

2^{κ}

equivalence classes of functions

κ \to κ

, so

∣ j (κ) ∣ \leq 2^{κ}

V_{λ}

. The target

M

regards

j (κ)

as measurable and contains

V_{κ + 1}

, so it sees

(2^{κ})^{M} < j (κ)

. Under the Generalized continuum hypothesis, this places

j (κ)

strictly above the ambient

κ^{+}

while its ambient cardinality is at most

κ^{+}

; hence it is not an ambient cardinal.

Strongly compact cardinal

 1  0

An uncountable cardinal

κ

is strongly compact when every

κ

-satisfiable theory in any

L_{κ, κ}

language is satisfiable, with no bound on the number of nonlogical symbols.

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Large cardinal by

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In set theory, a large cardinal is a type of cardinal number that possesses certain strong and often large-scale properties, which typically extend beyond the standard axioms of set theory (like Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC). Large cardinals are significant in the study of the foundations of mathematics because they often have implications for the consistency and structure of set theory. There are various kinds of large cardinals, each with different defining properties.

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Large cardinal

Supercompact cardinal ( $κ$ )

Supercompact Sigma-two downward reflection ( $V ⊨ σ ⟹ V_{κ} ⊨ σ (σ \in Σ_{2})$ )

Mahlo cardinal ( $κ Mahlo$ )

Mahloness is downward absolute to the constructible universe

Weakly inaccessible cardinal ( $cf (κ) = κ and κ is an uncountable limit cardinal$ )

Weakly Mahlo cardinal

Model of ZFC without weakly inaccessible cardinals

Strong limit cardinal

Singular strong limit power-set identity ( $2^{λ} = λ^{cf (λ)}$ )

Strongly inaccessible cardinal

Inaccessible limit of inaccessibles

Rank cutoff at the first inaccessible limit of inaccessibles ( $V_{ι} ⊨ ZFC + \infty IC + \neg ILI$ )

Keisler extension property

Worldly cardinal

Worldly cardinals below an inaccessible cardinal

Nested transitive models from a worldly cardinal

Arithmetic absoluteness for a rank-initial model

Kappa-satisfiable theory ( $κ$ -satisfiable)

Least-occurrence order on cardinal properties ( $<_{1}$ )

Nontransitivity of the least-occurrence order on cardinal properties

Weakly compact cardinal

Measurable cardinal

Omega-measurable cardinal ( $ω -measurable$ )

Least omega-measurable cardinal is measurable

Measurable cardinal is a strong limit cardinal

Measurable cardinal is one-strong

Two measurable cardinals under an ultrapower embedding

Moved critical point is not an ambient cardinal under GCH

Strongly compact cardinal

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