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Classifying Positive Equivalence Relations

Claudio Bernardi and Andrea Sorbi
The Journal of Symbolic Logic
Vol. 48, No. 3 (Sep., 1983), pp. 529-538
DOI: 10.2307/2273443
Stable URL: http://www.jstor.org/stable/2273443
Page Count: 10
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Classifying Positive Equivalence Relations
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Abstract

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, y ∈ ω, we have $x \sim_1 y \operatorname{iff} hx \sim_2 hy$. 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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