Ultrafilter translations. I: \((\lambda, \lambda)\)-compactness of logics with a cardinality quantifier
We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assuming \(V = L\), Gödel's Axiom of Constructibility, we prove that if \(\lambda > \omega_\alpha\) then the logic with the quantifier ``there exist \(\omega_\alpha\) many'' is \((\lambda, \lambda)\)-compact if and only if either \(\lambda\) is weakly compact or \(\lambda\) is singular of cofinality \(< \omega_{\alpha - 1}\). As a corollary, for every infinite cardinals \(\lambda\) and \(\mu\), there exists a \((\lambda, \lambda)\)-compact non-\((\mu, \mu)\)-compact logic if and only if either \(\lambda < \mu\) or \(\text{cf} \lambda < \text{cf} \mu\) or \(\lambda > \mu\) is weakly compact. Counterexamples are given showing that the above statements may fail, if \(V = L\) is not assumed. However, without special assumptions, analogous results are obtained for the stronger notion of \([\lambda, \lambda]\)-compactness.
scientific article; zbMATH DE number 851910
scientific article