Compactness Theorems for Infinitary Logics

ABSTRACT

We consider the logics L_ {kappa, lambda}, for kappa≥ lambda   infinite cardinals, and study possible generalizations to these logics of the usual compactness theorem for first-order logic. This will give rise to the definitions of weakly and strongly compact cardinals, and we will see their positions in the large cardinal hierarchy. We will also see some equivalent characterizations in terms of ultrafilters and compactness of topological spaces (generalizations of Tychonoff’s theorem).