Two Universal 3-quantifier Representations of Recursively Enumerable Sets

1. Let us agree on the following notation. Lower-case Latin letters from a to n (inclusively) with indices and without them will be used as variables for nonnegative integers, the remaining lower-case Latin letters will be used as variables for integers. Analogously, lower case Greek letters from to will be used as metavariables for nonnegative integers, and the rest of the Greek letters will be used as metavariables for integers. Upper case Latin letters will denote polynomials. Here as polynomials one means only polynomials with integer coeÆcients; this wont't be reminded to the reader below. 2. We say that a set R of nonnegative integers is represented by an arithmetic formula F with one free variable a if the equivalence a 2 R, F is true. As K.Godel showed, any recursively enumerable set is represented by some arithmetical formula. One can improve this result by putting various restrictions on the types of formulas. Such investigations were done in [2-13]. The aim of this paper is to show that every recursively enumerable set is represented by formulas of each of the two following types: