Journal of Symbolic Logic

Classifying Positive Equivalence Relations

Claudio Bernardi and Andrea Sorbi

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Abstract

Given two (positive) equivalence relations $\sim_1, \sim_2$ on the set $\omega$ of natural numbers, we say that $\sim_1$ is $m$-reducible to $\sim_2$ if there exists a total recursive function $h$ such that for every $x, y \in \omega$, we have $x \sim_1 y \operatorname{iff} hx \sim_2 hy$. We prove that the equivalence relation induced in $\omega$ 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 $\omega$ 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.

Article information

Source
J. Symbolic Logic, Volume 48, Issue 3 (1983), 529-538.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183741310

Mathematical Reviews number (MathSciNet)
MR716612

Zentralblatt MATH identifier
0528.03030

JSTOR
links.jstor.org

Citation

Bernardi, Claudio; Sorbi, Andrea. Classifying Positive Equivalence Relations. J. Symbolic Logic 48 (1983), no. 3, 529--538. https://projecteuclid.org/euclid.jsl/1183741310


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