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Duality between logics and equivalence relations


Author: Daniele Mundici
Journal: Trans. Amer. Math. Soc. 270 (1982), 111-129
MSC: Primary 03C95
DOI: https://doi.org/10.1090/S0002-9947-1982-0642332-1
MathSciNet review: 642332
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Abstract: Assuming $ \omega $ is the only measurable cardinal, we prove:

(i) Let $ \sim $ be an equivalence relation such that $ \sim \, = \,{ \equiv _L}$ for some logic $ L \leqslant {L^{\ast}}$ satisfying Robinson's consistency theorem (with $ {L^{\ast}}$ arbitrary); then there exists a strongest logic $ {L^ + } \leqslant {L^{\ast}}$ such that $ \sim \, = \,{ \equiv _{{L^ + }}}$; in addition, $ {L^ + }$ is countably compact if $ \sim \, \ne \, \cong $.

(ii) Let $ \dot \sim $ be an equivalence relation such that $ \sim \, = \,{ \equiv _{{L^0}}}$ for some logic $ {L^0}$ satisfying Robinson's consistency theorem and whose sentences of any type $ \tau $ are (up to equivalence) equinumerous with some cardinal $ {\kappa _\tau }$; then $ {L^0}$ is the unique logic $ L$ such that $ \sim \, = \,{ \equiv _L}$; furthermore, $ {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 $ { \equiv _L}$ for some compact logic $ L$ obeying Craig's interpolation theorem and whose sentences are equinumerous with some cardinal.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0642332-1
Article copyright: © Copyright 1982 American Mathematical Society

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