Pseudo-complements and ordinal logics based on consistency statements

1966 ◽  
Vol 31 (3) ◽  
pp. 359-364 ◽  
Author(s):  
Robert A. Di Paola
Keyword(s):  
Recursive Function ◽  
Peano Arithmetic ◽  
Additional Property ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Primitive Recursive

Following [1] we write {n} for the nth recursively enumerable (re) set; that is, {n} = {x|VyT(n, x, y)}. By a “pair (T, α)” we mean a consistent re extension T of Peano arithmetic P and an RE-formula α which numerates the non-logical axioms of T in P [4]. Given a pair (T, α) and a particular formula which binumerates the Kleene T predicate in P, there can be defined a primitive recursive function Nα such that and which has the additional property that {Nα(Nα(n))} = ø for all n.

On axiomatizability within a system

1953 ◽  
Vol 18 (1) ◽  
pp. 30-32 ◽  
Author(s):  
William Craig
Keyword(s):  
Recursive Function ◽  
Recursive Relation ◽  
Recursively Enumerable Set ◽  
Formal Systems ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Primitive Recursive ◽  
Recursive Set

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

Polynomially and superexponentially shorter proofs in fragments of arithmetic

1992 ◽  
Vol 57 (3) ◽  
pp. 844-863 ◽  
Author(s):  
Franco Montagna
Keyword(s):  
Fixed Points ◽  
Recursive Function ◽  
Peano Arithmetic ◽  
Modal Language ◽  
Primitive Recursive Function ◽  
Consistent Extension ◽  
Image Position ◽  
Primitive Recursive ◽  
Increasing Function

In Parikh [71] it is shown that, if T is an r.e. consistent extension of Peano arithmetic PA, then, for each primitive recursive function g, there is a formula φ of PA such that(In the following, Proof T(z, φ) and Prov T(φ) denote the metalinguistic assertions that z codes a proof of φ in T and that φ is provable in T respectively, where ProofT(z, ┌φ┐) and ProvT(┌φ┐) are the formalizations of Proof T(z,φ) and ProvT(φ) respectively in the language of PA, ┌φ┐ denotes the Gödel number of φ and ┌φ┐ denotes the corresponding numeral. Also, for typographical reasons, subscripts will not be made boldface.) If g is a rapidly increasing function, we express (1) by saying that ProvT(┌φ┐) has a much shorter proof modulo g than φ. Parikh's result is based on the fact that a suitable formula A(x), roughly asserting that (1) holds with x in place of φ, has only provable fixed points. In de Jongh and Montagna [89], this situation is generalized and investigated in a modal context. There, a characterization is given of arithmetical formulas arising from modal formulas of a suitable modal language which have only provable fixed points, and Parikh's result is obtained as a particular case.

Arithmetic Without the Exponential

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursive Function ◽  
Function Theory ◽  
Axiom System ◽  
Peano Arithmetic ◽  
Recursively Enumerable

§1. By an arithmetic term or formula, we mean a term or formula in which the exponential symbol E does not appear, and by an arithmetic relation (or set), we mean a relation (set) expressible by an arithmetic formula. By the axiom system P.A. (Peano Arithmetic), we mean the system P.E. with axiom schemes N10 and N11 deleted, and in the remaining axiom schemes, terms and formulas are understood to be arithmetic terms and formulas. The system P.A. is the more usual object of modern study (indeed, the system P.E. is rarely considered in the literature). We chose to give the incompleteness proof for P.E. first since it is the simpler. In this chapter, we will prove the incompleteness of P.A. and establish several other results that will be needed in later chapters. The incompleteness of P.A. will easily follow from the incompleteness of P.E., once we show that the relation xy = z is not only Arithmetic but arithmetic (definable from plus and times alone). We will first have to show that certain other relations are arithmetic, and as we are at it, we will show stronger results about these relations that will be needed, not for the incompleteness proof of this chapter, but for several chapters that follow—we will sooner or later need to show that certain key relations are not only arithmetic, but belong to a much narrower class of relations, the Σ1-relations, which we will shortly define. These relations are the same as those known as recursively enumerable. Before defining the Σ1-relations, we turn to a still narrower class, the Σ0-relations, that will play a key role in our later development of recursive function theory. §2. We now define the classes of Σ0-formulas and Σ0-relations and then the Σ1-formulas and relations. By an atomic Σ0-formula, we shall mean a formula of one of the four forms c1 + c2 = c3, c1 · c2 =c3, c1 = c2 or c1 ≤ c2, where each of c1, c2 or c3 is either a variable or a numeral (but some may be variables and others numerals).

Incompleteness along paths in progressions of theories

1962 ◽  
Vol 27 (4) ◽  
pp. 383-390 ◽  
Author(s):  
S. Feferman ◽  
C. Spector
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Order Number ◽  
Basic Logic ◽  
Natural Numbers ◽  
Logical Deduction ◽  
First Order ◽  
Primitive Recursive Function ◽  
Primitive Recursive ◽  
Recursive Definitions

We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

scholarly journals Non-primitive Recursive Function Definitions

1995 ◽  
Vol 2 (36) ◽  
Author(s):  
Sten Agerholm
Keyword(s):  
Recursive Function ◽  
Higher Order ◽  
Order Logic ◽  
Domain Theory ◽  
Higher Order Logic ◽  
Primitive Recursive Function ◽  
Primitive Recursive

This paper presents an approach to the problem of introducing<br />non-primitive recursive function definitions in higher order logic. A<br />recursive specification is translated into a domain theory version, where<br />the recursive calls are treated as potentially non-terminating. Once we<br />have proved termination, the original specification can be derived easily.<br />A collection of algorithms are presented which hide the domain theory<br />from a user. Hence, the derivation of a domain theory specification<br />has been automated completely, and for well-founded recursive function<br />specifications the process of deriving the original specification from the<br />domain theory one has been automated as well, though a user must<br />supply a well-founded relation and prove certain termination properties<br />of the specification. There are constructions for building well-founded<br />relations easily.

Torre models in the isols

1994 ◽  
Vol 59 (1) ◽  
pp. 140-150 ◽  
Author(s):  
Joseph Barback
Keyword(s):  
Number Theory ◽  
Structural Property ◽  
Recursive Function ◽  
Peano Arithmetic ◽  
The Other ◽  
Close Connection ◽  
New Variety ◽  
Nonstandard Model ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractIn [14] J. Hirschfeld established the close connection of models of the true AE sentences of Peano Arithmetic and homomorphic images of the semiring of recursive functions. This fragment of Arithmetic includes most of the familiar results of classical number theory. There are two nice ways that such models appear in the isols. One way was introduced by A. Nerode in [20] and is referred to in the literature as Nerode Semirings. The other way is called a tame model. It is very similar to a Nerode Semiring and was introduced in [6]. The model theoretic properties of Nerode Semirings and tame models have been widely studied by T. G. McLaughlin ([16], [17], and [18]).In this paper we introduce a new variety of tame model called a torre model. It has as a generator an infinite regressive isol with a nice structural property relative to recursively enumerable sets and their extensions to the isols. What is then obtained is a nonstandard model in the isols of the fragment of Peano Arithmetic with the following property: Let T be a torre model. Let f be any recursive function, and let fΛ be its extension to the isols. If there is an isol A with fΛ(A) ϵ T, then there is also an isol B ϵ T with fΛ(B) = fΛ(A).

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