Weakly compact cardinal

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In mathematics, a weakly compact cardinal is a certain kind of cardinal number; weakly compact cardinals are large cardinals, meaning that their existence can neither be proven nor disproven from the standard axioms of set theory.

Formally, a cardinal κ is weakly compact if and only if for every function f: κ 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, a subset S of κ is homogeneous for f if and only if either all of S×S maps to 0 or all of it maps to 1.

[edit] Equivalent formulations

The following are equivalent for any uncountable cardinal κ:

  1. κ is weakly compact.
  2. for every λ<κ, natural number n ≥ 2, and function f: κn → λ, there is a set of cardinality κ that is homogeneous for f.
  3. 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.
  4. Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
  5. κ is \Pi^1_1-indescribable.
  6. For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
  7. κ is κ-unfoldable.

[edit] See also

[edit] References

  • Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. 
  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3-540-00384-3. 
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