A note on Mathematics of infinity

1993 ◽  
Vol 58 (4) ◽  
pp. 1195-1200 ◽  
Author(s):  
Erik Palmgren
Keyword(s):  
Intuitionistic Logic ◽  
Type Theory ◽  
Natural Numbers ◽  
Disjunction Property ◽  
Recursive Functions ◽  
Choice Sequence ◽  
Heyting Arithmetic ◽  
Image Position

In the paper Mathematics of infinity, Martin-Löf extends his intuitionistic type theory with fixed “choice sequences”. The simplest, and most important instance, is given by adding the axiomsto the type of natural numbers. Martin-Löf's type theory can be regarded as an extension of Heyting arithmetic (HA). In this note we state and prove Martin-Löf's main result for this choice sequence, in the simpler setting of HA and other arithmetical theories based on intuitionistic logic (Theorem A). We also record some remarkable properties of the resulting systems; in general, these lack the disjunction property and may or may not have the explicit definability property. Moreover, they represent all recursive functions by terms.

A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property

1974 ◽  
Vol 39 (1) ◽  
pp. 67-78 ◽  
Author(s):  
D. M. Gabbay ◽  
D. H. J. De Jongh
Keyword(s):  
Intuitionistic Logic ◽  
Finite Model ◽  
Ordered Sets ◽  
Disjunction Property ◽  
University Of Illinois ◽  
Finite Model Property ◽  
Intermediate Logics ◽  
Heyting Arithmetic ◽  
Image Position ◽  
System 1

The intuitionistic propositional logic I has the following (disjunction) property.We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + (∼ϕ → (ψ ∨ α))→ ((∼ϕ→ψ) ∨ (∼ϕ→α)) and Scott's system I + ((∼ ∼ϕ→ϕ)→(ϕ ∨ ∼ϕ))→ (∼∼ϕ ∨ ∼ϕ). It was shown in [3c] that the first system has the finite-model property.In this note we shall construct a sequence of intermediate logics Dn with the following properties:These systems are presented both semantically and syntactically, using the remarkable correspondence between properties of partially ordered sets and axiom schemata of intuitionistic logic. This correspondence, apart from being interesting in itself (for giving geometric meaning to intuitionistic axioms), is also useful in giving independence proofs and obtaining proof theoretic results for intuitionistic systems (see for example, C. Smorynski, Thesis, University of Illinois, 1972, for independence and proof theoretic results in Heyting arithmetic).

The decidability of the Kreisel-Putnam system

1970 ◽  
Vol 35 (3) ◽  
pp. 431-437 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Finite Model ◽  
Model Property ◽  
Intuitionistic Propositional Logic ◽  
Disjunction Property ◽  
Finite Model Property ◽  
Image Position

The intuitionistic propositional logic I has the following disjunction property This property does not characterize intuitionistic logic. For example Kreisel and Putnam [5] showed that the extension of I with the axiomhas the disjunction property. Another known system with this propery is due to Scott [5], and is obtained by adding to I the following axiom:In the present paper we shall prove, using methods originally introduced by Segerberg [10], that the Kreisel-Putnam logic is decidable. In fact we shall show that it has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]. The decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.

A note on nominalistic syntax

1950 ◽  
Vol 14 (4) ◽  
pp. 226-227 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
Mathematical Theory ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Image Position ◽  
Positive Integers ◽  
Nominalistic Theory

In this note two independent comments are offered concerning nominalistic syntax.If one has a nominalistic theory of natural numbers (or positive integers) at one's disposal, one can of course readily formulate a nominalistic syntax by the familiar method of arithmetization. A nominalistic theory of natural numbers is formulated in a previous paper by the author, and thus a basis for a nominalistic syntax is already provided. (The objection that this theory is infinitistic would appear without force, because it provides the only known nominalistic basis for the theory of general recursive functions. On a finitistic basis one merely throws out this vital domain of mathematical theory.)The second comment is concerned with nominalistic syntax as formulated by Goodman and Quine. The nominalistic construction of natural numbers alluded to above owes much of its power to a primitive device of ancestral quantification. This device is somewhat more powerful than the notions Goodman and Quine allow themselves. Now nominalistic syntax in their somewhat narrower sense can be formulated by utilizing a relation akin to the relation L of the theory of ordered individuals developed elsewhere by the author. ‘L’ here is to designate the relation between inscriptions of being wholly to the left of.The Goodman-Quine primitive ‘C’, designating a relation of concatenation, and their primitive ‘Part’ with its field confined to inscriptions, can be defined in terms of ‘L’. Thus:.(This definition is essentially that of ‘P’ in O.I.)

A note on the representation of general recursive functions and the μ quantifier

1959 ◽  
Vol 55 (2) ◽  
pp. 145-148
Author(s):  
Alan Rose
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Image Position ◽  
Primitive Recursive

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schemasubject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

Transcendence of cardinals

1965 ◽  
Vol 30 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Power Set ◽  
Recursive Predicate

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

scholarly journals A class of harmonically convergent sets

1975 ◽  
Vol 20 (3) ◽  
pp. 301-304
Author(s):  
Torleiv Kløve
Keyword(s):  
Natural Numbers ◽  
Lower Case ◽  
Image Position

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

scholarly journals Computability and the algebra of fields: Some affine constructions

1980 ◽  
Vol 45 (1) ◽  
pp. 103-120 ◽  
Author(s):  
J. V. Tucker
Keyword(s):  
Algebraic Structure ◽  
Algebraic System ◽  
Finiteness Condition ◽  
Coordinate Systems ◽  
Natural Numbers ◽  
Algebraic Foundation ◽  
Image Position ◽  
Technical Idea ◽  
Natural Way ◽  
General Method

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagramwherein αr is r-fold α × … × α.This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

Universal recursion theoretic properties of r.e. preordered structures

1985 ◽  
Vol 50 (2) ◽  
pp. 397-406 ◽  
Author(s):  
Franco Montagna ◽  
Andrea Sorbi
Keyword(s):  
Equivalence Relation ◽  
Point Of View ◽  
Main Concern ◽  
Natural Numbers ◽  
Axiomatic Theory ◽  
Recursive Functions ◽  
Theoretic Point

When dealing with axiomatic theories from a recursion-theoretic point of view, the notion of r.e. preordering naturally arises. We agree that an r.e. preorder is a pair = 〈P, ≤P〉 such that P is an r.e. subset of the set of natural numbers (denoted by ω), ≤P is a preordering on P and the set {〈;x, y〉: x ≤Py} is r.e.. Indeed, if is an axiomatic theory, the provable implication of yields a preordering on the class of (Gödel numbers of) formulas of .Of course, if ≤P is a preordering on P, then it yields an equivalence relation ~P on P, by simply letting x ~Py iff x ≤Py and y ≤Px. Hence, in the case of P = ω, any preordering yields an equivalence relation on ω and consequently a numeration in the sense of [4]. It is also clear that any equivalence relation on ω (hence any numeration) can be regarded as a preordering on ω. In view of this connection, we sometimes apply to the theory of preorders some of the concepts from the theory of numerations (see also Eršov [6]).Our main concern will be in applications of these concepts to logic, in particular as regards sufficiently strong axiomatic theories (essentially the ones in which recursive functions are representable). From this point of view it seems to be of some interest to study some remarkable prelattices and Boolean prealgebras which arise from such theories. It turns out that these structures enjoy some rather surprising lattice-theoretic and universal recursion-theoretic properties.After making our main definitions in §1, we examine universal recursion-theoretic properties of some r.e. prelattices in §2.

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

On relations as coextensive with classes

1946 ◽  
Vol 11 (3) ◽  
pp. 71-72 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Natural Numbers ◽  
Previous Note ◽  
Image Position ◽  
Ordered Pairs

In a previous note I showed a new way to define the ordered pair. I made use of the notations ‘Nn’ (for class of natural numbers) and ‘Sv’ (for successor of v), remarking that they are readily defined without appeal to ordered pairs or relations. Adopting the auxiliary abbreviation: I defined the ordered pair thus:

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