Duality between logics and equivalence relations
Author:
Daniele Mundici
Journal:
Trans. Amer. Math. Soc. 270 (1982), 111129
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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 [CK]
 C. C. Chang and H. J. Keisler, Model theory, 2nd ed., NorthHolland, Amsterdam, 1977. MR 0532927 (58:27177)
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 S. Feferman, Two notes on abstract model theory. I, Fund. Math. 82 (1974), 153165.
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 J. Flum, Firstorder logic and its extensions, Lecture Notes in Math., vol. 499, SpringerVerlag, Berlin and New York, 1976, pp. 248310. MR 0401465 (53:5293)
 [Fr]
 H. Friedman, One hundred and two problems in mathematical logic, J. Symbolic Logic 40 (1975), 113129. MR 0369018 (51:5254)
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 P. Lindström, On extensions of elementary logic, Theoria 35 (1969), 111. MR 0244013 (39:5330)
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 J. D. Monk, Mathematical logic, SpringerVerlag, Berlin and New York, 1976. MR 0465767 (57:5656)
 [MS]
 J. A. Makowsky and S. Shelah, The theorems of Beth and Craig in abstract model theory. I: the abstract setting, Trans. Amer. Math. Soc. 256 (1979), 215239. MR 546916 (81b:03041)
 [MS1]
 , Positive results in abstract model theory (preprint).
 [MSS]
 J. A. Makowsky, S. Shelah and J. Stavi, logics and generalized quantifiers, Ann. Math. Logic 10 (1976), 155192. MR 0457146 (56:15362)
 [Mu1]
 D. Mundici, An algebraic result about soft model theoretical equivalence relations with an application to H. Friedman's fourth problem, J. Symbolic Logic 46 (1981), 523530. MR 627904 (82k:03056)
 [Mu2]
 , Robinson's consistency theorem in soft model theory, Trans. Amer. Math. Soc. 263 (1981), 231241. MR 590421 (82d:03063)
 [Mu3]
 , Compactness in any logic, Fund. Math. 116 (1982) (to appear).
 [Mu4]
 , Applications of manysorted Robinson consistency theorem, Z. Math. Logik Grundlagen Math. 27 (1981), 181188. MR 611855 (82k:03057)
 [Mu5]
 , Compactness, interpolation and Friedman's third problem (to appear).
 [Mu6]
 , Embeddings, amalgamation and elementary equivalence (to appear).
 [Na]
 M. E. Nadel, An arbitrary equivalence relation as elementary equivalence in abstract logic, Z. Math. Logik Grundlagen Math. 26 (1980), 103109. MR 564374 (82g:03070)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206423321
PII:
S 00029947(1982)06423321
Article copyright:
© Copyright 1982
American Mathematical Society
