Logics From Ultrafilters

Review of Symbolic Logic:1-18 (forthcoming)
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Abstract

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 ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega $ is $\omega $ -relatively compact iff some $D\in \Omega $ fails to be $\omega _1$ -complete iff ${\mathcal {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 ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$, if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $, then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

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Model Theory.C. C. Chang & H. Jerome Keisler - 1992 - Studia Logica 51 (1):154-155.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
On descendingly incomplete ultrafilters.Kenneth Kunen & Karel Prikry - 1971 - Journal of Symbolic Logic 36 (4):650-652.
A topological characterization of (λ, μ)*-compactness.Heikki Mannila - 1983 - Annals of Pure and Applied Logic 25 (3):301-305.

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