scholarly journals Monotone and 1–1 sets

1982 ◽  
Vol 33 (1) ◽  
pp. 62-75 ◽  
Author(s):  
D. B. Madan ◽  
R. W. Robinson
Keyword(s):  
Recursive Function ◽  
Infinite Subset ◽  
Recursively Enumerable Set ◽  
Recursively Enumerable

AbstractAn infinite subset of ω is monotone (1–1) if every recursive function is eventually monotone on it (eventually constant on it or eventually 1–1 on it). A recursively enumerable set is co-monotone (co-1–1) just if its complement is monotone (1–1). It is shown that no implications hold among the properties of being cohesive, monotone, or 1–1, though each implies r-cohesiveness and dense immunity. However it is also shown that co-monotone and co-1–1 are equivalent, that they are properly stronger than the conjunction of r-maximality and dense simplicity, and that they do not imply maximality.

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.

Degrees of formal systems

1958 ◽  
Vol 23 (4) ◽  
pp. 389-392 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Sufficient Conditions ◽  
Formal System ◽  
Independent Interest ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Axiomatizable Theory

In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of Gödel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable.In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A. (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.

scholarly journals Formalization of the MRDP Theorem in the Mizar System

2019 ◽  
Vol 27 (2) ◽  
pp. 209-221
Author(s):  
Karol Pąk
Keyword(s):  
Normal Form ◽  
Linear Equations ◽  
Negative Solution ◽  
Recursively Enumerable Set ◽  
Pell’S Equation ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Finite Products ◽  
Special Case ◽  
Martin Davis

Summary This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem. In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 − (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences $\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$ . We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9]. In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form. The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

Variations on promptly simple sets

1985 ◽  
Vol 50 (1) ◽  
pp. 138-148 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Recursive Function ◽  
Splitting Property ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets

In this paper we answer the question of whether all low sets with the splitting property are promptly simple. Further we try to make the role of lowness properties and prompt simplicity in the construction of automorphisms of the lattice of r.e. (recursively enumerable) sets more perspicuous. It turns out that two new properties of r.e. sets, which are dual to each other, are essential in this context: the prompt and the low shrinking property.In an earlier paper [4] we had shown (using Soare's automorphism construction [10] and [12]) that all r.e. generic sets are automorphic in the lattice ℰ of r.e. sets under inclusion. We called a set A promptly simple if Ā is infinite and there is a recursive enumeration of A and the r.e. sets (We)e∈N such that if We is infinite then there is some element (or equivalently: infinitely many elements) x of We such that x gets into A “promptly” after its appearance in We (i.e. for some fixed total recursive function f we have x ∈ Af(s), where s is the stage at which x entered We). Prompt simplicity in combination with lowness turned out to capture those properties of r.e. generic sets that were used in the mentioned automorphism result. In a following paper with Shore and Stob [7] we studied an ℰ-definable consequence of prompt simplicity: the splitting property.

TERMINAL WEIGHTED L-SYSTEMS

1990 ◽  
Vol 04 (01) ◽  
pp. 95-112 ◽  
Author(s):  
NALINAKSHI NIRMAL ◽  
R. RAMA
Keyword(s):  
Regular Matrix ◽  
Recursively Enumerable Set ◽  
Recursively Enumerable ◽  
The Family ◽  
L Systems

Terminal weights are attached to L-systems by replacing each terminal generated by an OL-system by fa(i) in the ith step of a derivation. The family of terminal weighted OL languages will be equal to the recursively enumerable set. Terminal weights are attached to EOL-regular matrix languages and also to OL array languages. Parquet deformations are generated by TWEOL-RMS.

Σ2-collection and the infinite injury priority method

1988 ◽  
Vol 53 (1) ◽  
pp. 212-221 ◽  
Author(s):  
Michael E. Mytilinaios ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degree ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Recursively Enumerable ◽  
Standard Application ◽  
Priority Method ◽  
Order Arithmetic ◽  
Bounded Fragment

AbstractWe show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P−) together with Σ2-collection (BΣ2). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither lown nor highn-1, while true, cannot be established in P− + BΣn+1. Consequently, no bounded fragment of first order arithmetic establishes the facts that the highn and lown jump hierarchies are proper on the recursively enumerable degrees.

Post's problem and his hypersimple set

1973 ◽  
Vol 38 (3) ◽  
pp. 446-452 ◽  
Author(s):  
Carl G. Jockusch ◽  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Recursively Enumerable ◽  
Standard Enumeration

A standard enumeration of the recursively enumerable (r.e.) sets is an acceptable numbering {Wn}n∈N of the r.e. sets in the sense of Rogers [5, p. 41], together with a 1:1 recursive function f with range In his quest for nonrecursive incomplete r.e. sets Post [4] constructed a hypersimple set Hf, relative to a fixed but unspecified standard enumeration f. Although it was later shown that hyper-simplicity does not guarantee incompleteness, the ironic possibility remained that Post's own particular hypersimple set might be incomplete. We settle the question by proving that H, may be either complete or incomplete depending upon which standard enumeration f is used. In contrast, D. A. Martin has shown [3] that Post's simple set S [4, p. 298] is complete for any standard enumeration. Furthermore, what most modern recursion theorists would regard as the “natural” construction of a hypersimple set (which we give in §1) is also complete for any standard enumeration.There are two conclusions to be drawn from these results. First, they substantiate the often repeated remark among recursion theorists that Post's hypersimple set construction is a precursor of priority constructions because priorities play a strong role, and because there is a great deal of “restraint” which tends to keep elements out of the set. Secondly, the results warn recursion theorists that more properties than might have been supposed depend upon which standard enumeration is chosen at the beginning of the construction of some r.e. set.

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).

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