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Classifying positive equivalence relations

  • Claudio Bernardi (a1) and Andrea Sorbi (a1)

Given two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, yω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.

From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

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[1] C. Bernardi , On the relation provable equivalence and on partitions in effectively inseparable sets, Studia Logica, vol. 40 (1981), pp. 2937.

[2] J.P. Cleave , Creative functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 205212.

[3] Ju. L. Eršov , Theorie der Numerierungen. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 19 (1973), pp. 289388.

[4] Ju. L. Eršov , Theorie der Numerierungen. II, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 473584.

[5] Ju. L. Eršov , Positive equivalences (English translation), Algebra and Logic, vol. 10 (1973), pp. 378394.

[6] J. Myhill , Creative sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 97108.

[7] M.B. Pour-el and H. Putnam , Recursively enumerable classes and their application to recursive sequences of formal theories, Archiv für Mathematische Logik und Grundlagenforschung, vol. 8 (1965), pp. 104121.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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