Worldly cardinal

A cardinal

κ

is worldly when

V_{κ} ⊨ ZFC

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

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Worldly cardinals below an inaccessible cardinal

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

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

If

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.

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Worldly cardinal by

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The term "worldly cardinal" isn't widely recognized in common discourse or established literature, so it could refer to different concepts depending on context. However, it seems to suggest two distinct meanings: 1. **Religious Context**: In a more traditional sense, a "cardinal" typically refers to a high-ranking official in the Roman Catholic Church, a cardinal is a member of the clergy who is appointed by the Pope and is eligible to participate in papal elections.

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