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 - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, the least weakly compact is still smaller than a lot of other large cardinals, such as totally reflecting cardinals.