Logics from ultrafilters

Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class Omega of uniform ultrafilters generates a Delta-closed logic L-Omega. L-Omega is omega-relatively compact iff some D is an element of Omega fails to be omega(1)-complete iff L-Omega does not contain the quantifier "there are uncountably many." If Omega is a set, or if it contains a countably incomplete ultrafilter, then L-Omega is not generated by Mostowski cardinality quantifiers. Assuming 0(# ) or L-mu, if D is an element of Omega is a uniform ultrafilter over a regular cardinal nu, then every family Psi of formulas in L-Omega with |Phi| <= nu |Phi| <= satisfies the compactness theorem. In particular, if Omega is a proper class of uniform ultrafilters over regular cardinals, L-Omega is compact.