In mathematical set theory, a worldly cardinal is a
cardinal
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of cardinal in the family Cardinalidae
**''Cardinalis cardinalis'', or northern cardinal, the ...
κ such that the
rank ''V''
κ is a
model of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
.
Relationship to inaccessible cardinals
By
Zermelo's theorem on
inaccessible cardinals, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that (''V''
κ, ''V''
κ+1) is a model of second order Zermelo-Fraenkel set theory.
Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a
singular cardinal
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, s ...
.
[Kanamori (2003), Lemma 6.1, p. 57.]
The following are in strictly increasing order, where ι is the least inaccessible cardinal:
* The least worldly κ.
* The least worldly κ and λ (κ<λ, and same below) with ''V''
κ and ''V''
λ satisfying the same theory.
* The least worldly κ that is a limit of worldly cardinals (equivalently, a limit of κ worldly cardinals).
* The least worldly κ and λ with ''V''
κ ≺
Σ2 ''V''
λ (this is higher than even a κ-fold iteration of the above item).
* The least worldly κ and λ with ''V''
κ ≺ ''V''
λ.
* The least worldly κ of cofinality ω
1 (corresponds to the extension of the above item to a chain of length ω
1).
* The least worldly κ of cofinality ω
2 (and so on).
* The least κ>ω with ''V''
κ satisfying replacement for the language augmented with the (''V''
κ,∈) satisfaction relation.
* The least κ inaccessible in ''L''
κ(''V''
κ); equivalently, the least κ>ω with ''V''
κ satisfying replacement for formulas in ''V''
κ in the
infinitary logic ''L''
∞,ω.
* The least κ with a transitive model M⊂''V''
κ+1 extending ''V''
κ satisfying
Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
.
* (not a worldly cardinal) The least κ with ''V''
κ having the same Σ
2 theory as ''V''
ι.
* The least κ with ''V''
κ and ''V''
ι having the same theory.
* The least κ with ''L''
κ(''V''
κ) and ''L''
ι(''V''
ι) having the same theory.
* (not a worldly cardinal) The least κ with ''V''
κ and ''V''
ι having the same Σ
2 theory with real parameters.
* (not a worldly cardinal) The least κ with ''V''
κ ≺
Σ2 ''V''
ι.
* The least κ with ''V''
κ ≺ ''V''
ι.
* The least infinite κ with ''V''
κ and ''V''
ι satisfying the same ''L''
∞,ω statements that are in ''V''
κ.
* The least κ with a transitive model M⊂''V''
κ+1 extending ''V''
κ and satisfying the same sentences with parameters in ''V''
κ as ''V''
ι+1 does.
* The least inaccessible cardinal ι.
References
*
*
External links
Worldly cardinalin Cantor's attic
Large cardinals
{{settheory-stub