Duality between logics and equivalence relations

Abstract

Assuming ω \omega is the only measurable cardinal, we prove: (i) Let ∼ \sim be an equivalence relation such that ∼ = ≡ L \sim \, = \,{ \equiv _L} for some logic L ⩽ L ∗ L \leqslant {L^{\ast }} satisfying Robinson’s consistency theorem (with L ∗ {L^{\ast }} arbitrary); then there exists a strongest logic L + ⩽ L ∗ {L^ + } \leqslant {L^{\ast }} such that ∼ = ≡ L + \sim \, = \,{ \equiv _{{L^ + }}} ; in addition, L + {L^ + } is countably compact if ∼ ≠ ≅ \sim \, \ne \, \cong . (ii) Let ∼ ˙ \dot \sim be an equivalence relation such that ∼ = ≡ L 0 \sim \, = \,{ \equiv _{{L^0}}} for some logic L 0 {L^0} satisfying Robinson’s consistency theorem and whose sentences of any type τ \tau are (up to equivalence) equinumerous with some cardinal κ τ {\kappa _\tau } ; then L 0 {L^0} is the unique logic L L such that ∼ = ≡ L \sim \, = \,{ \equiv _L} ; furthermore, L 0 {L^0} is compact and obeys Craig’s interpolation theorem. We finally give an algebraic characterization of those equivalence relations ∼ \sim which are equal to ≡ L { \equiv _L} for some compact logic L L obeying Craig’s interpolation theorem and whose sentences are equinumerous with some cardinal.