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Transactions of the American Mathematical Society

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

 
 

 

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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Article copyright: © Copyright 1982 American Mathematical Society