Duality between logics and equivalence relations

Author:
Daniele Mundici

Journal:
Trans. Amer. Math. Soc. **270** (1982), 111-129

MSC:
Primary 03C95

MathSciNet review:
642332

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Abstract: Assuming is the only measurable cardinal, we prove:

(i) Let be an equivalence relation such that for some logic satisfying Robinson's consistency theorem (with arbitrary); then there exists a strongest logic such that ; in addition, is countably compact if .

(ii) Let be an equivalence relation such that for some logic satisfying Robinson's consistency theorem and whose sentences of any type are (up to equivalence) equinumerous with some cardinal ; then is the unique logic such that ; furthermore, is compact and obeys Craig's interpolation theorem.

We finally give an algebraic characterization of those equivalence relations which are equal to for some compact logic obeying Craig's interpolation theorem and whose sentences are equinumerous with some cardinal.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0642332-1

Article copyright:
© Copyright 1982
American Mathematical Society