Weakly Compact Cardinal
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a weakly compact cardinal is a certain kind of
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
introduced by ; 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'': º 2 → there is a set of
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
κ that is homogeneous for ''f''. In this context, º 2 means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
either all of 'S''sup>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

The following are equivalent for any uncountable cardinal κ: # κ is weakly compact. # for every λ<κ,
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
n ≥ 2, and function f: ºsup>n → λ, there is a set of cardinality κ that is homogeneous for f. # κ is inaccessible and has the tree property, that is, every
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...
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 \Pi^1_1- indescribable. # κ has the extension property. In other words, for all ''U'' ⊂ ''V''κ there exists a transitive set ''X'' with κ ∈ ''X'', and a subset ''S'' ⊂ ''X'', such that (''V''κ, ∈, ''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 ''L''κ,κ satisfies the weak compactness theorem. # κ is inaccessible and the infinitary language ''L''κ,ω satisfies the weak compactness theorem. # κ is inaccessible and for every
transitive set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions holds: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. S ...
M of cardinality κ with κ \in M, ^M\subset M, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding j from M to a transitive set N of cardinality κ such that ^N\subset N, with critical point crit(j)=κ. # \kappa=\kappa^ (\kappa^ defined as \sum_\kappa^\lambda) and every \kappa-complete filter of a \kappa-complete field of sets of cardinality \leq\kappa is contained in a \kappa-complete ultrafilter. (W. W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', p.185) # \kappa has Alexander's property, i.e. for any space X with a \kappa-subbase \mathcal A with cardinality \leq\kappa, and every cover of X by elements of \mathcal A has a subcover of cardinality <\kappa, then X is \kappa-compact. (W. W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', p.182--185) # (2^)_\kappa is \kappa-compact. (W. W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', p.185) A language ''L''κ,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

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 \kappa is weakly compact, then there are chains of well-founded elementary end-extensions of (V_\kappa,\in) of arbitrary length <\kappa^+.p.6 Weakly compact cardinals remain weakly compact in L. Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.Bagaria, Magidor, Mancilla
On the Consistency Strength of Hyperstationarity
p.3. (2019)


See also

* List of large cardinal properties


References

* * * *


Citations

{{reflist Large cardinals