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.

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 On the General Erdős-Turán Conjecture

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Georges Grekos ◽  
Labib Haddad ◽  
Charles Helou ◽  
Jukka Pihko
Keyword(s):  
Finite Sequence ◽  
General Term ◽  
Natural Numbers ◽  
Intrinsic Interest ◽  
Positive Integers ◽  
Increasing Function

The general Erdős-Turán conjecture states that if A is an infinite, strictly increasing sequence of natural numbers whose general term satisfies an≤cn2, for some constant c>0 and for all n, then the number of representations functions of A is unbounded. Here, we introduce the function ψ(n), giving the minimum of the maximal number of representations of a finite sequence A={ak:1≤k≤n} of n natural numbers satisfying ak≤k2 for all k. We show that ψ(n) is an increasing function of n and that the general Erdős-Turán conjecture is equivalent to limn→∞ψ(n)=∞. We also compute some values of ψ(n). We further introduce and study the notion of capacity, which is related to the ψ function by the fact that limn→∞ψ(n) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.

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.

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)).

Patterns of projecta

1981 ◽  
Vol 46 (2) ◽  
pp. 287-295
Author(s):  
Adam Krawczyk
Keyword(s):  
Finite Sequence ◽  
Natural Numbers ◽  
Main Technique

AbstractRoughly speaking, a pattern is a finite sequence coding the set of natural numbers n for which the Σn+1 projectum is less than the Σn projectum for a given admissible ordinal. We prove that for each pattern there exists an ordinal realizing it. Several results on the orderings of patterns are given. We conclude the paper with remarks on Δn projecta. The main technique, used throughout the paper, is Jensen's Uniformisation Theorem.

Robustness of the Algorithm of Identification of the Type of Dynamic Object Found at the Finite Sequence of 2D Background Frames of the Optoelectron Device

2019 ◽  
Vol 40 (12) ◽  
pp. 2062-2076
Author(s):  
A. N. Katulev ◽  
A. N. Sotnikov ◽  
V. K. Kemaykin ◽  
I. V. Kozhukhin
Keyword(s):  
Finite Sequence ◽  
Dynamic Object

The Argument from (Natural) Numbers

2018 ◽  
Author(s):  
Tyron Goldschmidt
Keyword(s):  
Natural Numbers ◽  
Existence Of God ◽  
Divine Attribute ◽  
Rule Out

This chapter considers Plantinga’s argument from numbers for the existence of God. Plantinga sees divine psychologism as having advantages over both human psychologism and Platonism. The chapter begins with Plantinga’s description of the argument, including the relation of numbers to any divine attribute. It then argues that human psychologism can be ruled out completely. However, what rules it out might rule out divine psychologism too. It also argues that the main problem with Platonism might also be a problem with divine psychologism. However, it will, at the least, be less of a problem. In any case, there are alternative, possibly viable views about the nature of numbers that have not been touched by Plantinga’s argument. In addition, the chapter touches on the argument from properties, and its relation to the argument from numbers.

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

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