An existence theorem for recursion categories

1990 ◽  
Vol 55 (3) ◽  
pp. 1252-1268 ◽  
Author(s):  
Alex Heller
Keyword(s):  
Number Theory ◽  
Existence Theorem ◽  
Free Semigroup ◽  
Recursion Theory ◽  
Traditional Theory ◽  
Ordinal Numbers ◽  
Partial Recursive Functions ◽  
Uniform Generation ◽  
Similar Development ◽  
Proper Generalization

Recursion theory has been dominated by one example. The notion of a dominical category was introduced (see [2] and [3]) at least in part in the hope of subverting this dominance. The structures of number theory, freed from this original context, led to much of modern algebra with a consequent enrichment of number theory itself: it seemed not unreasonable to attempt to contribute, however modestly, to a similar development of recursion theory. Thus (in the loosest sort of analogy) recursion categories are intended to stand in relation to the class of partial recursive functions as rings do to the rational integers.But if such a generalization is to serve its intended purpose, it ought to allow examples substantially different in spirit from the prototype. Recursion categories are, of course, the proper generalization of classical recursion theory. As a novice in this formidable field I am incapable of encompassing its full extent and must rely on the advice of others (I thank in particular R. DiPaola) in concluding that most of these earlier generalizations are not of this character. Indeed, the examples of recursion categories so far adduced (cf., in addition to the references above, [1] and [4]) are themselves closely tied to the classical one.We attempt here to open the door to radically disparate examples by proving an existence theorem allowing the construction of recursion categories in a wide, and so far unexplored, variety of contexts very distant from that of the natural or ordinal numbers and their subsets, the locus of the traditional theory.In order to do this, it has been necessary to introduce a number of unfamiliar notions—unfamiliar, that is, even in the context of the earlier discussion of dominical categories. This novelty is perhaps mitigated by the fact that, however unfamiliar, they cannot be said to be unusual, as the discussion later will show. We mention in particular those of a “formally free semigroup”, which retains the indexation function of a free semigroup, while giving up the characteristic universal property and that of “uniform generation”, which generalizes finite generation in the fashion assorted to our argument.

Admissible ordinals and intrinsic consistency

1970 ◽  
Vol 35 (3) ◽  
pp. 389-400 ◽  
Author(s):  
Michael Machtey
Keyword(s):  
Analogous Result ◽  
Fundamental Problem ◽  
Recursion Theory ◽  
Infinite Cardinal ◽  
Reduction Procedure ◽  
Partial Recursive Function ◽  
Ordinal Numbers ◽  
Partial Recursive Functions ◽  
Uniform Manner ◽  
Admissible Ordinals

This paper deals with a fundamental problem in the theory of recursion on initial segments of the ordinal numbers which was originated by Kripke [4] and Platek [8]. Following Kreisel-Sacks [3], we define a notion of relative recursiveness for this abstract recursion theory which allows us to speak of a function ƒ being weakly recursive in a set A via a partial recursive function g; this makes it reasonable to speak of partial recursive functions as ‘reduction procedures’. We say that a reduction procedure g is intrinsically consistent if for any set A there is a partial function ƒ which is weakly recursive in A via g. It is an elementary result of ordinary recursion theory that an arbitrary reduction procedure can be replaced in a uniform manner by an intrinsically consistent reduction procedure without the loss of any function f being reducible to any set A. In §3 we show that for recursion theory on the ordinals less than any infinite cardinal the analogous result is consistent with the axioms of set theory. In §5 we show that for all other cases of abstract recursion theory the analogous result fails in a very strong way to be true.

Definability of r. e. sets in a class of recursion theoretic structures

1983 ◽  
Vol 48 (3) ◽  
pp. 662-669 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursion Theory ◽  
Universal Function ◽  
Independent Interest ◽  
Suitable Model ◽  
Natural Numbers ◽  
Construction Techniques ◽  
Partial Recursive Functions ◽  
Priority Method ◽  
Recursive Isomorphism ◽  
Direct Attack

It is known [4, Theorem 11-X(b)] that there is only one acceptable universal function up to recursive isomorphism. It follows from this that sets definable in terms of a universal function alone are specified uniquely up to recursive isomorphism. (An example is the set K, which consists of all n such that {n}(n) is defined, where λn, m{n}(m) is an acceptable universal function.) Many of the interesting sets constructed and studied by recursion theorists, however, have definitions which involve additional notions, such as a specific enumeration of the graph of a universal function. In particular, many of these definitions make use of the interplay between the purely number-theoretic properties of indices of partial recursive functions and their purely recursion-theoretic properties.This paper concerns r.e. sets that can be defined using only a universal function and some purely number-theoretic concepts. In particular, we would like to know when certain recursion-theoretic properties of r.e. sets definable in this way are independent of the particular choice of universal function (equivalently, independent of the particular way in which godel numbers are identified with natural numbers).We will first develop a suitable model-theoretic framework for discussing this question. This will enable us to classify the formulas defining r.e. sets by their logical complexity. (We use the number of alternations of quantifiers in the prenex form of a formula as a measure of logical complexity.) We will then be able to examine the question at each level.This work is an approach to the question of when the recursion-theoretic properties of an r.e. set are independent of the particular parameters used in its construction. As such, it does not apply directly to the construction techniques most commonly used at this time for defining particular r.e. sets, e.g., the priority method. A more direct attack on this question for these techniques is represented by such works as [3] and [5]. However, the present work should be of independent interest to the logician interested in recursion theory.

The hereditary partial effective functionals and recursion theory in higher types

1984 ◽  
Vol 49 (4) ◽  
pp. 1319-1332 ◽  
Author(s):  
G. Longo ◽  
E. Moggi
Keyword(s):  
Recursion Theory ◽  
Type Structure ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Higher Types ◽  
Elementary Technique

AbstractA type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings).By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (HEO) are easily characterized.

Minimal α-recursion theoretic degrees

1973 ◽  
Vol 38 (1) ◽  
pp. 18-28 ◽  
Author(s):  
John M. MacIntyre
Keyword(s):  
Recursive Function ◽  
Regular Cardinal ◽  
Recursion Theory ◽  
Minimal Degree ◽  
The Other ◽  
Ordinal Numbers ◽  
Definition Of

This paper investigates the problem of extending the recursion theoretic construction of a minimal degree to the Kripke [2]-Platek [5] recursion theory on the ordinals less than an admissible ordinal α, a theory derived from the Takeuti [11] notion of a recursive function on the ordinal numbers. As noted in Sacks [7] when one generalizes the recursion theoretic definition of relative recursiveness to α-recursion theory for α > ω the two usual definitions give rise to two different notions of reducibility. We will show that whenever α is either a countable admissible or a regular cardinal of the constructible universe there is a subset of α whose degree is minimal for both notions of reducibility. The result is an excellent example of a theorem of ordinary recursion theory obtainable via two different constructions, one of which generalizes, the other of which does not. The construction which cannot be lifted to α-recursion theory is that of Spector [10]. We sketch the reasons for this in §3.

Interpretations of set theory and ordinal number theory

1967 ◽  
Vol 32 (2) ◽  
pp. 145-161
Author(s):  
Mariko Yasugi
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Ordinal Number ◽  
Recursive Relation ◽  
Ordinal Numbers ◽  
Primitive Recursive

In [3], Takeuti developed the theory of ordinal numbers (ON) and constructed a model of Zermelo-Fraenkel set theory (ZF), using the primitive recursive relation ∈ of ordinal numbers. He proved:(1) If A is a ZF-provable formula, then its interpretation A0 in ON is ON-provable;(2) Let B be a sentence of ordinal number theory. Then B is a theorem of ON if and only if the natural translation B* of B in set theory is a theorem of ZF;(3) (V = L)° holds in ON.

TRUNCATION AND SEMI-DECIDABILITY NOTIONS IN APPLICATIVE THEORIES

2018 ◽  
Vol 83 (3) ◽  
pp. 967-990
Author(s):  
GERHARD JÄGER ◽  
TIMOTEJ ROSEBROCK ◽  
SATO KENTARO
Keyword(s):  
Recursion Theory ◽  
Natural Numbers ◽  
First Order ◽  
Recursive Functions ◽  
Partial Recursive Functions

AbstractBON+ is an applicative theory and closely related to the first order parts of the standard systems of explicit mathematics. As such it is also a natural framework for abstract computations. In this article we analyze this aspect of BON+ more closely. First a point is made for introducing a new operation τN, called truncation, to obtain a natural formalization of partial recursive functions in our applicative framework. Then we introduce the operational versions of a series of notions that are all equivalent to semi-decidability in ordinary recursion theory on the natural numbers, and study their mutual relationships over BON+ with τN.

Formal development of ordinal number theory

1955 ◽  
Vol 20 (2) ◽  
pp. 95-104
Author(s):  
Steven Orey
Keyword(s):  
Number Theory ◽  
Ordinal Number ◽  
Ordered Sets ◽  
Stringent Condition ◽  
Easy Consequence ◽  
Basic Properties ◽  
Ordinal Numbers ◽  
Order Types ◽  
Proper Subclass ◽  
Definition Of

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

Non-standard models for formal logics

1950 ◽  
Vol 15 (2) ◽  
pp. 113-129 ◽  
Author(s):  
J. Barkley Rosser ◽  
Hao Wang
Keyword(s):  
Number Theory ◽  
Standard Model ◽  
Formal Logic ◽  
Precise Definition ◽  
Ordinal Numbers ◽  
Equality Relation ◽  
Standard Models ◽  
Definition Of ◽  
Positive Integers

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex (for instance, if it is adequate for number theory), then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either:(a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model.(b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦.(c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦.

Recursion theory and the lambda-calculus

1982 ◽  
Vol 47 (1) ◽  
pp. 67-83 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Lambda Calculus ◽  
Recursion Theory ◽  
Natural Class ◽  
Recursive Functions ◽  
Partial Recursive Functions

AbstractA semantics for the lambda-calculus due to Friedman is used to describe a large and natural class of categorical recursion-theoretic notions. It is shown that if e1 and e2 are gödel numbers for partial recursive functions in two standard ω-URS's which both act like the same closed lambda-term, then there is an isomorphism of the two ω-URS's which carries e1 to e2.

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