In this paper we prove that the preordering . of provable implication over any recursively enumerable theory T containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function F for .. A recursive function F is a density function if it computes, for A and B with A ' B, an element C such that A ' C ' B. The function is extensional if it preserves T -provable equivalence. Secondly, we prove a general result that implies that, for extensions of elementary arithmetic, the ordering . restricted to † n -sentences is uniformly dense. In the last section we provide historical notes and background material.
The paper contains some observations on e-complete, precomplete, and uniformly finitely precomplete r. e. equivalence relations. Among these are a construction of a uniformly finitely precomplete r.e. equivalence which is neither e-nor precomplete, an extension of Lachlan's theorem that all precomplete r. e. equivalences are isomorphic, and a characterization of sets of fixed points of endomorphisms of uniformly finitely precomplete r. e. equivalences.
Mathematics Subject Classification: 03D45.A positive equivalence S is an r.e. equivalence relation on the set of non-negative integers w . Occasionally, especially when more than one equivalence is in sight, we shall be stipulating that each positive equivalence is an equivalence relation on its own private copy of w , so that we think of S as a pair (DomS,-s), D o m S being, essentially, w . Thus, a morphism p : S -7 from a positive equivalence S to a positive equivalence 7 is a mapping p : DomS/-s-Dom'T/-7 for which there exists a total recursive function h : D o m S -D o m T such that p ( [ z ] s ) = [ h ( z ) ] i for all t E DomS, where [D]R stands for the closure of an element (a set of elements, a list of elements etc.) D of DomR under -R. The recursive function h is then said to represent p. Clearly, p can as well be represented by an r.e. subset H of D o m S x D o m T such that for every 2 E DomS there is a pair (I, w ) E H such that z -S 2 and p ( [ z ] s ) = [WIT, for from (an index of) such an H one can effectively construct a representation of p in the form of a total recursive function.Positive equivalences together with the morphisms just described constitute a category equivalent to the category of positively numerated sets introduced in ERSOV [6, Kapitel 11, 31. Our paper focuses around its precomplete objects.'11 would like to thank SEFUKZHAN ACYBAEVICH BADAEV for helpful correspondence and ALBERT
')Current addrecrs:A positive equivalence is called precompleie if for any recursive program we can effectively compute a number such that if this program converges, then its output is in the same equivalence class as that number. Precomplete (positive) equivalences have been extensively studied since the time of ERSOV [5].An application-motivated yet natural generalization of this notion is that of a uniformly finitely precomplete (u.f.p.) positive equivalence introduced by MONTAGNA [8].Here one requires the same but under the condition that the program can only output a number in one of the finite number of equivalence classes specified beforehand. (Precise definitions are given in Section 1.) This being a proper generalization, there are u.f.p. positive equivalences that are not precomplete. Such are e. g. the e-complete positive equivalences (see BERNARDI and MONTAGNA [3], LACHLAN [7]), an example of which is the provable equivalence among sentences of formalized arithmetic (see BERNARDI [2]).The aim of this paper is to document several miscellaneous observations on these three classes of equivalences. Section 1 introduces the necessary definitions an...
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