Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Independent Gödel sentences and independent sets

1975 ◽  
Vol 40 (2) ◽  
pp. 159-166
Author(s):  
A. M. Dawes ◽  
J. B. Florence
Keyword(s):  
Positive Integer ◽  
Independent Sets ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Simple Set ◽  
Logical Notion ◽  
Infinite Set ◽  
Element Set

In this paper we investigate some of the recursion-theoretic problems which are suggested by the logical notion of independence.A set S of natural numbers will be said to be k-independent (respectively, ∞-independent) if, roughly speaking, in every correct system there is a k-element set (respectively, an infinite set) of independent true sentences of the form x ∈ S. S will be said to be effectively independent (respectively, absolutely independent) if given any correct system we can generate an infinite set of independent (respectively, absolutely independent) true sentences of the form x ∈ S.We prove that(a) S is absolutely independent ⇔S is effectively independent ⇔S is productive;(b) for every positive integer k there is a Π1 set which is k-independent but not (k + 1)-independent;(c) there is a Π1 set which is k-independent for all k but not ∞-independent;(d) there is a co-simple set which is ∞-independent.We also give two new proofs of the theorem of Myhill [1] on the existence of an infinite set of Σ1 sentences which are absolutely independent relative to Peano arithmetic. The first proof uses the existence of an absolutely independent Π1 set of natural numbers, and the second uses a modification of the method of Gödel and Rosser.

Mathematical Knowledge and the Origin of Phenomenology

2021 ◽  
Vol 21 ◽  
pp. 273-294
Author(s):  
Gabriele Baratelli ◽  
Keyword(s):  
Mathematical Knowledge ◽  
Cardinal Number ◽  
The Other ◽  
Mathematical Science ◽  
Natural Numbers ◽  
Symbolic Thinking ◽  
The One

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall.

scholarly journals Between 2- and 3-Colorability

10.37236/4673 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Positive Integer ◽  
Random Graphs ◽  
The Other ◽  
Chromatic Numbers ◽  
Other Hand ◽  
Odd Cycles

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n$, $1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$.  These results imply the existence of random graphs with circular chromatic numbers $\chi_c$ satisfying $2<\chi_c(G)<2+\delta$ for arbitrarily small $\delta$, and also that $2.5\leq \chi_c(G_{n,\frac 4 n})<3$ w.h.p.

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Proceed with care, because some infinities are larger than others

2020 ◽  
pp. 281-292
Author(s):  
Susan D'Agostino
Keyword(s):  
Number Line ◽  
Mathematics Students ◽  
Natural Numbers ◽  
Real Numbers ◽  
Countable Infinity ◽  
One To One ◽  
Infinite Set

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

A Characterization of the Topological Dimension

1982 ◽  
Vol 25 (4) ◽  
pp. 487-490
Author(s):  
Gerd Rodé
Keyword(s):  
Hausdorff Space ◽  
The Other ◽  
Natural Numbers ◽  
Topological Dimension ◽  
Other Hand ◽  
The One

AbstractThis paper gives a new characterization of the dimension of a normal Hausdorff space, which joins together the Eilenberg-Otto characterization and the characterization by finite coverings. The link is furnished by the notion of a system of faces of a certain type (N1,..., NK), where N1,..., NK, K are natural numbers. It is shown that a space X contains a system of faces of type (N1,..., NK) if and only if dim(X) ≥ N1 + … + NK. The two limit cases of the theorem, namely Nk = 1 for 1 ≤ k ≤ K on the one hand, and K = 1 on the other hand, give the two known results mentioned above.

A pseudohedron

2004 ◽  
Vol 88 (512) ◽  
pp. 226-229
Author(s):  
H. Martyn Cundy
Keyword(s):  
The Other ◽  
Regular Polygons ◽  
Spherical Trigonometry ◽  
Equilateral Triangles ◽  
Infinite Set

The other day I received a long tube from a Canadian stranger containing a large poster featuring over a hundred polyhedra, including ‘all 92 Johnson polyhedra’. This term, though probably unfamiliar this side of the pond, was not completely unknown to me; it means convex polyhedra, excluding the regular and Archimedean ones, all of whose faces are regular polygons. Of course, as usual, we have to exclude those naughty polyhedra whose faces go around in pairs collecting squares (prisms) or equilateral triangles (antiprisms) and don’t know when to stop. The word convex is vital, otherwise there would be another infinite set. A lot of them are rather trivial, like sticking pyramids on the faces of a dodecahedron, but they include the deltahedra and many other interesting members. But they have at least one imitator who didn't quite make the grade. Trying to discover why, and how to coach him so that he would, I found that my spherical trigonometry was getting rather rusty so I set out to make one and see what was happening. I thought perhaps other readers would like to share this piece of antiresearch.

scholarly journals On radicals of infinite matrix rings

1969 ◽  
Vol 16 (3) ◽  
pp. 195-203 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Infinite Matrix ◽  
Natural Numbers ◽  
Index Set ◽  
Matrix Rings ◽  
Infinite Set

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Tiwadee Musunthia ◽  
Jörg Koppitz
Keyword(s):  
Natural Numbers ◽  
Maximal Subsemigroups ◽  
Countably Infinite ◽  
Infinite Set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

scholarly journals II. Note on legendre’s coefficients

1878 ◽  
Vol 27 (185-189) ◽  
pp. 381-383
Keyword(s):  
Positive Integer ◽  
Royal Society ◽  
The Other ◽  
Definite Integral ◽  
Other Hand

In the “Proceedings of the Royal Society,” vol. xxvii, pp. 63-71, Professor Adams has given, by an inductive process, the development of the product of any two of Legendre’s Coefficients in a series of the Coefficients; from this is immediately deduced the value of the integral between the limits -1 and +1 of the product of any three of the Coefficients. On the other hand, if we know the value of this definite integral, we can immediately deduce the development of the product of any two of the Coefficients. Thus it may be of interest to give a brief investigation of the value of the definite integral. I follow the notation adopted by Professor Adams. The formula to be established is ∫ 1 -1 P m P n P p dμ = 2/2 s + 1 A ( s - m ) A ( s - n ) A ( s - p ) / A( s ), where 2 s = m + n + P , and the functional symbol A( r ) is thus defined: if r is a positive integer A ( r ) = 1. 3. 5. . (2 r - 1) /1. 2. 3. . r , and in all other cases A ( r ) is to be considered zero, except when r = 0, and then it is to be considered = 1.

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