In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ1, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by Švejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree (...) structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result. (shrink)

We present a decision procedure for the ∀∃ theory of the lattice of Σ1 sentences of Peano Arithmetic.

We study effectively inseparable (abbreviated as e.i.) prelattices (i.e., structures of the form$L = \langle \omega, \wedge, \vee,0,1,{ \le _L}\rangle$whereωdenotes the set of natural numbers and the following four conditions hold: (1)$\wedge, \vee$are binary computable operations; (2)${ \le _L}$is a computably enumerable preordering relation, with$0{ \le _L}x{ \le _L}1$for everyx; (3) the equivalence relation${ \equiv _L}$originated by${ \le _L}$is a congruence onLsuch that the corresponding quotient structure is a nontrivial bounded lattice; (4) the${ \equiv _L}$-equivalence classes of 0 and 1 (...) form an effectively inseparable pair of sets). Solving a problem in (Montagna & Sorbi, 1985) we show (Theorem 4.2), that ifLis an e.i. prelattice then${ \le _L}$is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relationRthere exists a computable functionfreducingRto${ \le _L}$, i.e.,$xRy$if and only if$f\left( x \right){ \le _L}f\left( y \right)$, for all$x,y$. In fact (Corollary 5.3)${ \le _L}$is locally universal, i.e., for every pair$a{ < _L}b$and every c.e. preordering relationRone can find a reducing functionffromRto${ \le _L}$such that the range offis contained in the interval$\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also (Theorem 5.7)${ \le _L}$is uniformly dense, i.e., there exists a computable functionfsuch that for every$a,b$if$a{ < _L}b$then$a{ < _L}f\left( {a,b} \right){ < _L}b$, and if$a{ \equiv _L}a\prime$and$b{ \equiv _L}b\prime$then$f\left( {a,b} \right){ \equiv _L}f\left( {a\prime,b\prime } \right)$. Some consequences and applications of these results are discussed: in particular (Corollary 7.2) for$n \ge 1$the c.e. preordering relation on${{\rm{\Sigma }}_n}$sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s systemRorQis locally universal and uniformly dense; and (Corollary 7.3) the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)