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).

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

Π10 classes and Boolean combinations of recursively enumerable sets

1974 ◽  
Vol 39 (1) ◽  
pp. 95-96 ◽  
Author(s):  
Carl G. Jockusch
Keyword(s):  
Turing Machine ◽  
Peano Arithmetic ◽  
Binary Sequences ◽  
First Order ◽  
Recursive Tree ◽  
Recursively Enumerable ◽  
Private Correspondence ◽  
Recursively Enumerable Sets ◽  
Binary Input

Let be the collection of all sets which are finite Boolean combinations of recursively enumerable (r.e.) sets of numbers. Dale Myers asked [private correspondence] whether there exists a nonempty class of sets containing no member of . In this note we construct such a class. The motivation for Myers' question was his observation (reported in [1]) that the existence of such a class is equivalent to the assertion that there is a finite consistent set of tiles which has no m-trial tiling of the plane for any m (obeying the “origin constraint”). (For explanations of these terms and further results on tilings of the plane, cf. [1] and [5].) In addition to the application to tilings, the proof of our results gives some information on bi-immune sets and on complete extensions of first-order Peano arithmetic.A class of sets may be roughly described as the class of infinite binary input tapes for which a fixed Turing machine fails to halt, or alternatively as the class of infinite branches of a recursive tree of finite binary sequences. (In these definitions, sets of numbers are identified with the corresponding binary sequences.) Precise definitions, as well as many results concerning such classes, may be found in [3] and [4].

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.

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).

Remarks on Peano Arithmetic

2000 ◽  
Vol 20 (1) ◽  
Author(s):  
Charles Sayward
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Peano Arithmetic ◽  
The Other ◽  
Natural Numbers ◽  
Recursive Definition ◽  
Explicit Definition

<p>Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is a point about set theory, not number theory. We have learned more about the distinction between logic and set theory than was known in Russell's day, especially as this affects logicist aspirations.</p>

Recursively enumerable sets which are uniform for finite extensions

1971 ◽  
Vol 36 (2) ◽  
pp. 271-287 ◽  
Author(s):  
Donald A. Alton
Keyword(s):  
Recursive Function ◽  
Background Information ◽  
Partial Recursive Function ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

Let W0, W1 … be one of the usual enumerations of recursively enumerable (r.e.) subsets of the set N of nonnegative integers. (Background information will be given later.) Suggestions of Anil Nerode led to the followingDefinitions. Let B be a subset of N and let ψ be a partial recursive function.

On complexity properties of recursively enumerable sets

1973 ◽  
Vol 38 (4) ◽  
pp. 579-593 ◽  
Author(s):  
M. Blum ◽  
I. Marques
Keyword(s):  
Complexity Theory ◽  
Recursive Function ◽  
Strong Sense ◽  
Important Goal ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions ◽  
Do So

An important goal of complexity theory, as we see it, is to characterize those partial recursive functions and recursively enumerable sets having some given complexity properties, and to do so in terms which do not involve the notion of complexity.As a contribution to this goal, we provide characterizations of the effectively speedable, speedable and levelable [2] sets in purely recursive theoretic terms. We introduce the notion of subcreativeness and show that every program for computing a partial recursive function f can be effectively speeded up on infinitely many integers if and only if the graph of f is subcreative.In addition, in order to cast some light on the concepts of effectively speedable, speedable and levelable sets we show that all maximal sets are levelable (and hence speedable) but not effectively speedable and we exhibit a set which is not levelable in a very strong sense but yet is effectively speedable.

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.

Three Special Topics

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursive Function ◽  
General Reader ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Recursive Set

The topics of this chapter are of more specialized interest and are not necessary for the results of our final chapter. They will probably be of more interest to the specialist (particularly the results of Section III) than to the general reader. We know from Chapter 2 that if all recursive sets are representable in S, then S is undecidable. We also know from Chapter 4 that if all recursively enumerable sets are representable in S, then S is not only undecidable but generative. We might also ask the question, If all recursive sets are representable in S, is S necessarily generative? Shoenfield [1961] answered this question negatively. He constructed an axiomatizable system in which all recursive sets are representable, and so the system is undecidable, but he showed that the system is not creative. Let us say that all recursive sets are uniformly representable in S if there is a recursive function g(x) such that for any number i, if ωi is a recursive set, then g(i) is the Godel number of a formula which represents ωi in S. We will show that if all recursive sets are uniformly representable in S, then S is generative. We also showed in Chapter 2 that if all recursive sets are definable in S and S is consistent, then the pair (P,R) of its nuclei is recursively inseparable. Under the same hypothesis, is the pair (P,R) necessarily effectively inseparable? The answer is no. In Shoenfield’s system all recursive sets are not only representable, but definable. However, the set P is not creative. Hence the pair (P, R) of nuclei of the system, though recursively inseparable, is not effectively inseparable.

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