Henryk Greniewski. Functors of the propositional calculus. VI Zjazd Matematyków Polskich, Warszawa 20–23 IX 1948, supplement to Annales de la Société Polonaise de Mathématique, vol. 22, Cracow1950, pp. 78–86. - Henryk Greniewski. Certain notions of the theory of numbers as applied to the propositional calculus. English with brief Polish summary. Časopis pro pěstováni matematiky a fysiky, vol. 74 (1950), pp. 132–136. - Henryk Greniewski. Groups and fields definable in the propositional calculus. Towarzystwo Naukowe Warszawskie, Sprawozdania z posiedzé wydzialu III nauk matematyczno fizycznych (Socété des Sciences et des Lettres de Varsovle, Comptes-rendus des séances de la classe III sciences mathématiques et physiques), vol. 43 (for 1950, pub. 1952), pp. 53–48. - H. Greniewski. Arithmetics of natural numbers as part of the bi-valued propositional calculus. Colloquium matkematicum, vol. 2 no. 3–4 (for 1951, pub. 1952), pp. 291–297.

1968 ◽  
Vol 33 (2) ◽  
pp. 304-305
Author(s):  
G. T. Kneebone
Keyword(s):  
Propositional Calculus ◽  
Natural Numbers ◽  
Theory Of Numbers

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.

Arithmetics of natural numbers as part of the bi-valued propositional calculus

1951 ◽  
Vol 2 (3-4) ◽  
pp. 291-297 ◽  
Author(s):  
H. Greniewski
Keyword(s):  
Propositional Calculus ◽  
Natural Numbers

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.

m-valued sub-system of (m + n)-valued propositional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 177-181 ◽  
Author(s):  
Tzu-Hua Hoo
Keyword(s):  
Propositional Calculus ◽  
Natural Numbers ◽  
Class Membership ◽  
Present Treatment ◽  
Product Operation ◽  
Image Position ◽  
Function Of Two Variables

An (m + n)-valued propositional calculus2 may happen to be a subsystem of an m-valued propositional calculus , though the converse is never true. This fact may give us the impression that, as m grows, the content of becomes meagre. The present treatment is intended to remove this impression by constructing a complete, m-valued sub-system of any (m + n)-valued propositional calculus.In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.2.1 = df stands for definational identity in the syntactic language.2.11 ≡ stands for definational identity in the object calculi.2.2 ∊, ⊂, ∩, {x1, …, xn} are used in their meanings as customarily employed in the theory of sets—∊ for class membership, ⊂ for proper inclusion, ∩ for the product operation of classes, {x1, …, xn} for the class with x1, …, xn as its only elements.2.3 x, y, z are used as unspecified natural numbers including 0. m, n, i, j are used as unspecified natural numbers other than 0.2.401 Definition. δ = df as the function of two variables defined for any x, y such that2.41 Definition. For m ≧ 2, ιm = df the function of two variables denned on the set {0, …, m − 1} such that ιm (x, y) = y − x for x ≦ y and ιn(x, y) = 0 for x > y.

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.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

Asymptotics with remainder term for moments of the total cycle number of random A-permutation

2021 ◽  
Vol 31 (1) ◽  
pp. 51-60
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Remainder Term ◽  
Random Permutation ◽  
Natural Numbers ◽  
Cycle Number ◽  
Fixed Set ◽  
Cycle Lengths

Abstract Dedicated to the memory of Alexander Ivanovich Pavlov. We consider the set of n-permutations with cycle lengths belonging to some fixed set A of natural numbers (so-called A-permutations). Let random permutation τ n be uniformly distributed on this set. For some class of sets A we find the asymptotics with remainder term for moments of total cycle number of τ n .

The elliptic products of Jacobi and the theory of linear congruences

1926 ◽  
Vol 23 (4) ◽  
pp. 337-355
Author(s):  
P. A. MacMahon
Keyword(s):  
Elliptic Functions ◽  
Linear Congruences ◽  
Image Position ◽  
Theory Of Numbers

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobiare of great importance.

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