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LOGICS FROM ULTRAFILTERS
- Part of
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- Journal:
- The Review of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 28 November 2023, pp. 1-18
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- Article
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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.
