Paradox of the class of all grounded classes

1953 ◽  
Vol 18 (2) ◽  
pp. 114-114 ◽  
Author(s):  
Shen Yuting
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Image Position ◽  
Special Case

A class A for which there is an infinite progression of classes A1, A2, … (not necessarily all distinct) such thatis said to be groundless. A class which is not groundless is said to be grounded. Let K be the class of all grounded classes.Let us assume that K is a groundless class. Then there is an infinite progression of classes A1, A2, … such thatSince A1 ϵ K, A1 is a grounded class; sinceA1 is also a groundless class. But this is impossible.Therefore K is a grounded class. Hence K ϵ K, and we haveTherefore K is also a groundless class.This paradox forms a sort of triplet with the paradox of the class of all non-circular classes and the paradox of the class of all classes which are not n-circular (n a given natural number). The last of the three includes as a special case the paradox of the class of all classes which are not members of themselves (n = 1).More exactly, a class A1 is circular if there exists some positive integer n and classes A2, A3, …, An such thatFor any given positive integer n, a class A1 is n-circular if there are classes A2, …, An, such thatQuite obviously, by arguments similar to the above, we get a paradox of the class of all non-circular classes and a paradox of the class of all classes which are not n-circular, for each positive integer n.

The þ-function in λ-K-conversion

1937 ◽  
Vol 2 (4) ◽  
pp. 164-164 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Definition Of ◽  
Image Position

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.

On the Inequality

1974 ◽  
Vol 17 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Pal Fischer
Keyword(s):  
Positive Integer ◽  
Renyi’S Entropy ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Special Case

In this article, we are concerned with the following inequality(1)where 0<pi<1, 0<q<1, (i=l, 2,…,n), n is a fixed positive integer, n≥2 and f(p)≠0 for <p<l.This inequality was first considered by A. Renyi, who gave the general differentiate solution of (1) for n≥3, [1]. With the help of this inequality one can characterize Renyi’s entropy [2].We shall state later the Renyi’s result, which will be a special case of the Theorem 3.

scholarly journals Some infinite Fibonacci groups

1975 ◽  
Vol 19 (3) ◽  
pp. 311-314 ◽  
Author(s):  
D. L. Johnson
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Free Group ◽  
Factor Group ◽  
Image Position ◽  
Special Case

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

scholarly journals On the Inequality

1979 ◽  
Vol 22 (4) ◽  
pp. 483-489 ◽  
Author(s):  
Peter Kardos
Keyword(s):  
Positive Integer ◽  
Functional Inequality ◽  
Renyi’S Entropy ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Special Case

In this paper, we are concerned with the functional inequality1where 0 < Pi < l, 0 < qi < l, fi(p)≠0, for 0 < P < 1, (i = 1, 2,..., n) and n is a fixed positive integer, n ≥ 2.Inequality (1) was studied by Rényi and Fischer, (see [1], [3]) in the special case2and this provided a characterization of Rényi's entropy.

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

scholarly journals The AND/OR Theorem for Perceptrons

1992 ◽  
Vol 53 (1) ◽  
pp. 1-8
Author(s):  
I. D. Bruce ◽  
D. Easdown
Keyword(s):  
Positive Integer ◽  
Special Case

AbstractMinsky and Papert claim that, for any positive integer n, there exist predicates of order 1 whose conjunction and disjunction have order greater than n. Their proof is amended and a stronger result obtained of which their claim is a special case.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

Sign in / Sign up

Export Citation Format

Share Document