Weakly compact cardinal
A weakly compact cardinal is a certain kind of large cardinal. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\):
- \(\kappa\) is 0-Ramsey.
- \(\kappa\) is \(\Pi^1_1\)-indescribable.
- \(\kappa\) is \(\kappa\)-unfoldable.
- The partition property \(\kappa \to (\kappa)^2_2\) holds.
Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem.
The existence of a weakly compact cardinal is not provable in ZFC, assuming its existence - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, since "\(\kappa\) is weakly compact" is a \(\Pi^1_2\) property of \(V_\kappa\),[1] i.e. a \(\Pi_2\) property of \(V_{\kappa+1}\),[2] a totally reflecting cardinal, or even a \(\Pi^1_2\)-indescribable cardinal, is larger than the least weakly compact cardinal.
Note that, unlike the relation between weakly and strongly inaccessible cardinals, and weakly and strongly Mahlo cardinals, strongly compact cardinals are always significantly greater than weakly compact cardinals, both in terms of consistency strength and size. Also, any weakly compact cardinal is necessarily a strong limit, and there is no known weakening which allows \(2^{\aleph_0}\) to be weakly compact, unlike the case with weakly inaccessible and weakly inaccessible cardinals.
References[edit | edit source]
- ↑ Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:10.1007/978-3-540-88867-3_2. ISBN 3-540-00384-3
- ↑ J. D. Hamkins, "Local properties in set theory" (2014), blog post. Accessed 29 August 2023.