Some formal relative consistency proofs

1953 ◽  
Vol 18 (2) ◽  
pp. 136-144 ◽  
Author(s):  
Robert McNaughton
Keyword(s):  
Number Theory ◽  
Natural Number ◽  
Natural Numbers ◽  
Quantification Theory ◽  
Usual Manner ◽  
Function Calculus ◽  
Image Position ◽  
Positive Integers

These systems are roughly natural number theory in, respectively, nth order function calculus, for all positive integers n. Each of these systems is expressed in the notation of the theory of types, having variables with type subscripts from 1 to n. Variables of type 1 stand for natural numbers, variables of type 2 stand for classes of natural numbers, etc. Primitive atomic wff's (well-formed formulas) of Tn are those of number theory in variables of type 1, and of the following kind for n > 1: xi ϵ yi+1. Other wff's are formed by truth functions and quantifiers in the usual manner. Quantification theory holds for all the variables of Tn. Tn has the axioms Z1 to Z9, which are, respectively, the nine axioms and axiom schemata for the system Z (natural number theory) on p. 371 of [1]. These axioms and axiom schemata contain only variables of type 1, except for the schemata Z2 and Z9, which are as follows:where ‘F(x1)’ can be any wff of Tn. Identity is primitive for variables of type 1; for variables of other types it is defined as follows:

A finite arithmetic

1973 ◽  
Vol 38 (2) ◽  
pp. 232-248 ◽  
Author(s):  
Philip T. Shepard
Keyword(s):  
Number Theory ◽  
Natural Number ◽  
Singular Term ◽  
New Method ◽  
Axiomatic System ◽  
Natural Numbers ◽  
Substantial Loss ◽  
First Order ◽  
Finite Arithmetic

In this paper I shall argue that the presumption of infinitude may be excised from the area of mathematics known as natural number theory with no substantial loss. Except for a few concluding remarks, I shall restrict my concern in here arguing the thesis to the business of constructing and developing a first-order axiomatic system for arithmetic (called ‘FA’ for finite arithmetic) that contains no theorem to the effect that there are infinitely many numbers.The paper will consist of five parts. Part I characterizes the underlying logic of FA. In part II ordering of natural numbers is developed from a restricted set of axioms, induction schemata are proved and a way of expressing finitude presented. A full set of axioms are used in part III to prove working theorems on comparison of size. In part IV an ordinal expression is defined and characteristic theorems proved. Theorems for addition and multiplication are derived in part V from definitions in terms of the ordinal expression of part IV. The crucial final constructions of part V present a new method of replacing recursive characterizations by strict definitions.In view of our resolution not to assume that there are infinitely many numbers, we shall have to deal with the situation where singular arithmetic terms of FA may fail to refer. For I know of no acceptable and systematic way of avoiding such situations. As a further result, singular-term instances of universal generalizations of FA are not to be inferred directly from the generalizations themselves. Nevertheless, (i) (x)(y)(x + y = y + x), for example, and all its instances are provable in FA.

scholarly journals A variance method in combinatorial number theory

1969 ◽  
Vol 10 (2) ◽  
pp. 126-129 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Number Theory ◽  
Variance Method ◽  
Combinatorial Number Theory ◽  
Prime Factors ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

Let s = s(a1, a2,...., ar) denote the number of integer solutions of the equationsubject to the conditionsthe ai being given positive integers, and square brackets denoting the integral part. Clearly s (a1,..., ar) is also the number s = s(m) of divisors of which contain exactly λ prime factors counted according to multiplicity, and is therefore, as is proved in [1], the cardinality of the largest possible set of divisors of m, no one of which divides another.

On a multi-type critical age-dependent branching process

1970 ◽  
Vol 7 (3) ◽  
pp. 523-543 ◽  
Author(s):  
H. J. Weiner
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Continuous Distribution ◽  
Cell Type ◽  
Distinguishable Particle ◽  
Age Dependent ◽  
Critical Age ◽  
Image Position

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi(t), Gi(0+) = 0. At the end of its life, cell i is replaced by j1 new cells of type 1, j2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

Repercussions of a Problem of Erdős and Ulam on Density Ideals

1990 ◽  
Vol 42 (5) ◽  
pp. 902-914 ◽  
Author(s):  
Winfried Just
Keyword(s):  
Boolean Algebra ◽  
Natural Number ◽  
Natural Numbers ◽  
Logarithmic Density ◽  
Density Zero ◽  
Image Position ◽  
The Ideal

By P(ω) we denote the Boolean algebra of all subsets of the set ω of natural numbers. We identify each natural number with the set of its predecessors and define: the ideal of sets of density zero, and the ideal of sets of logarithmic density zero.

On a multi-type critical age-dependent branching process

1970 ◽  
Vol 7 (03) ◽  
pp. 523-543 ◽  
Author(s):  
H. J. Weiner
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Continuous Distribution ◽  
Cell Type ◽  
Distinguishable Particle ◽  
Age Dependent ◽  
Critical Age ◽  
Image Position

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi (t), Gi (0+) = 0. At the end of its life, cell i is replaced by j 1 new cells of type 1, j 2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

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.

Type two partial degrees

1978 ◽  
Vol 43 (4) ◽  
pp. 623-629
Author(s):  
Ko-Wei Lih
Keyword(s):  
Recursive Function ◽  
Natural Extension ◽  
Degree Theory ◽  
Recursion Theory ◽  
Natural Numbers ◽  
Partial Functions ◽  
Reduction Procedure ◽  
Turing Reducibility

Roughly speaking partial degrees are equivalence classes of partial objects under a certain notion of relative recursiveness. To make this notion precise we have to state explicitly (1) what these partial objects are; (2) how to define a suitable reduction procedure. For example, when the type of these objects is restricted to one, we may include all possible partial functions from natural numbers to natural numbers as basic objects and the reduction procedure could be enumeration, weak Turing, or Turing reducibility as expounded in Sasso [4]. As we climb up the ladder of types, we see that the usual definitions of relative recursiveness, equivalent in the context of type-1 total objects and functions, may be extended to partial objects and functions in quite different ways. First such generalization was initiated by Kleene [2]. He considers partial functions with total objects as arguments. However his theory suffers the lack of transitivity, i.e. we may not obtain a recursive function when we substitute a recursive function into a recursive function. Although Kleene's theory provides a nice background for the study of total higher type objects, it would be unsatisfactory when partial higher type objects are being investigated. In this paper we choose the hierarchy of hereditarily consistent objects over ω as our universe of discourse so that Sasso's objects are exactly those at the type-1 level. Following Kleene's fashion we define relative recursiveness via schemes and indices. Yet in our theory, substitution will preserve recursiveness, which makes a degree theory of partial higher type objects possible. The final result will be a natural extension of Sasso's Turing reducibility. Due to the abstract nature of these objects we do not know much about their behaviour except at the very low types. Here we pay our attention mainly to type-2 objects. In §2 we formulate basic notions and give an outline of our recursion theory of partial higher type objects. In §3 we introduce the definitions of singular degrees and ω-consistent degrees which are two important classes of type-2 objects that we are most interested in.

Note on the representation of integers as sums of certain powers

1969 ◽  
Vol 65 (2) ◽  
pp. 445-446 ◽  
Author(s):  
K. Thanigasalam
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Representation Of Integers ◽  
Image Position ◽  
Large Natural Number

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

On Arithmetic Convolutions

1966 ◽  
Vol 9 (3) ◽  
pp. 287-296 ◽  
Author(s):  
T.M. K. Davison
Keyword(s):  
Complex Numbers ◽  
Natural Numbers ◽  
Image Position ◽  
Dirichlet Product ◽  
Positive Integers

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given bywhere the summation condition means sum over all positive integers d which divide n.

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