Weakly compact cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.)
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1.
The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
Equivalent formulations
[edit]The following are equivalent for any uncountable cardinal κ:
- κ is weakly compact.
- for every λ<κ, natural number n ≥ 2, and function f: [κ]n → λ, there is a set of cardinality κ that is homogeneous for f. (Drake 1974, chapter 7 theorem 3.5)
- κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ.
- Every linear order of cardinality κ has an ascending or a descending sequence of order type κ. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
- κ is -indescribable.
- κ has the extension property. In other words, for all U ⊂ Vκ there exists a transitive set X with κ ∈ X, and a subset S ⊂ X, such that (Vκ, ∈, U) is an elementary substructure of (X, ∈, S). Here, U and S are regarded as unary predicates.
- For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
- κ is κ-unfoldable.
- κ is inaccessible and the infinitary language Lκ,κ satisfies the weak compactness theorem.
- κ is inaccessible and the infinitary language Lκ,ω satisfies the weak compactness theorem.
- κ is inaccessible and for every transitive set of cardinality κ with κ , , and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding from to a transitive set of cardinality κ such that , with critical point κ. (Hauser 1991, Theorem 1.3)
- κ is a strongly inaccessible ramifiable cardinal. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
- ( defined as ) and every -complete filter of a -complete field of sets of cardinality is contained in a -complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
- has Alexander's property, i.e. for any space with a -subbase with cardinality , and every cover of by elements of has a subcover of cardinality , then is -compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
- is -compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
A language Lκ,κ is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.
Properties
[edit]Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
If is weakly compact, then there are chains of well-founded elementary end-extensions of of arbitrary length .[1]p.6
Weakly compact cardinals remain weakly compact in .[2] Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.[3]
See also
[edit]References
[edit]- Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier Science Ltd, ISBN 0-444-10535-2
- Erdős, Paul; Tarski, Alfred (1961), "On some problems involving inaccessible cardinals", Essays on the foundations of mathematics, Jerusalem: Magnes Press, Hebrew Univ., pp. 50–82, MR 0167422
- Hauser, Kai (1991), "Indescribable Cardinals and Elementary Embeddings", Journal of Symbolic Logic, 56 (2), Association for Symbolic Logic: 439–457, doi:10.2307/2274692, JSTOR 2274692, S2CID 288779
- Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3
Citations
[edit]- ^ Villaveces, Andres (1996). "Chains of End Elementary Extensions of Models of Set Theory". arXiv:math/9611209.
- ^ T. Jech, 'Set Theory: The third millennium edition' (2003)
- ^ Bagaria, Magidor, Mancilla. On the Consistency Strength of Hyperstationarity, p.3. (2019)