An extensional variety of extended basic logic

1958 ◽  
Vol 23 (1) ◽  
pp. 13-21 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Mathematical Analysis ◽  
Symbolic Logic ◽  
Basic Logic ◽  
Natural Numbers ◽  
Strong Principle ◽  
Theory Of Numbers

The system K′ of “extended basic logic” lacks a principle of extensionality. In this paper a system KE′ will be presented which is like K′ in many respects but which does possess a fairly strong principle of extensionality by way of rule 6.37 below. It will be shown that KE′ is free from contradiction. KE′ is especially well suited for formalizing the theory of numbers presented in my paper, On natural numbers, integers, and rationals. The methods used there can be applied even more directly here because of the freedom of KE′ from type restrictions, but the details of such a derivation of a portion of mathematics will not be presented in this paper. It is evident, moreover, that KE′ contains at least as much of mathematical analysis as does K′, and perhaps considerably more. The method of carrying out proofs in KE′ is closely similar to that used in my book Symbolic logic, and could be expressed in similar notation.

Note on an idea of Fitch

1949 ◽  
Vol 14 (3) ◽  
pp. 175-176 ◽  
Author(s):  
John R. Myhill
Keyword(s):  
Finite Sequence ◽  
Basic Logic ◽  
Natural Numbers ◽  
Recursive Predicate

The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:Every recursive predicate is recursively definite. If R(x1, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x1, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.

On Waring's problem for cubes

1991 ◽  
Vol 109 (2) ◽  
pp. 229-256 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Singular Series ◽  
Natural Numbers ◽  
Additive Theory ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Quantitative Form ◽  
Image Position ◽  
Theory Of Numbers

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

A problem in additive number theory

1988 ◽  
Vol 103 (1) ◽  
pp. 27-33 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Number Theory ◽  
Additive Number Theory ◽  
Interesting Problem ◽  
Natural Numbers ◽  
Additive Theory ◽  
Additive Number ◽  
Image Position ◽  
Theory Of Numbers

The determination of the minimal s such that all large natural numbers n admit a representation asis an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.

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)).

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