# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Duality between logics and equivalence relationsHTML articles powered by AMS MathViewer

by Daniele Mundici
Trans. Amer. Math. Soc. 270 (1982), 111-129 Request permission

## 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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