A strongly minimal expansion of (ω, s)

1987 ◽  
Vol 52 (1) ◽  
pp. 205-207
Author(s):  
David Marker
Keyword(s):  
Natural Numbers ◽  
Image Position

We will show that there is a nontrivial strongly minimal expansion of (ω, s), the natural numbers with successor. Pillay and Steinhorn [1] proved that there is no -minimal expansion of (ω, ≤). This result provides an interesting contrast.The strongly minimal expansion of (ω, s) is very easy to describe. Consider the order-two permutation of ω, π recursively defined byLet T be Th(ω, s, π, 0).

scholarly journals A class of harmonically convergent sets

1975 ◽  
Vol 20 (3) ◽  
pp. 301-304
Author(s):  
Torleiv Kløve
Keyword(s):  
Natural Numbers ◽  
Lower Case ◽  
Image Position

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

scholarly journals Computability and the algebra of fields: Some affine constructions

1980 ◽  
Vol 45 (1) ◽  
pp. 103-120 ◽  
Author(s):  
J. V. Tucker
Keyword(s):  
Algebraic Structure ◽  
Algebraic System ◽  
Finiteness Condition ◽  
Coordinate Systems ◽  
Natural Numbers ◽  
Algebraic Foundation ◽  
Image Position ◽  
Technical Idea ◽  
Natural Way ◽  
General Method

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagramwherein αr is r-fold α × … × α.This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

On relations as coextensive with classes

1946 ◽  
Vol 11 (3) ◽  
pp. 71-72 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Natural Numbers ◽  
Previous Note ◽  
Image Position ◽  
Ordered Pairs

In a previous note I showed a new way to define the ordered pair. I made use of the notations ‘Nn’ (for class of natural numbers) and ‘Sv’ (for successor of v), remarking that they are readily defined without appeal to ordered pairs or relations. Adopting the auxiliary abbreviation: I defined the ordered pair thus:

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

scholarly journals Note on Continued Fractions and the Sequence of Natural Numbers

1932 ◽  
Vol 27 ◽  
pp. ix-xiii ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Continued Fractions ◽  
Natural Question ◽  
Natural Numbers ◽  
Image Position

When a student first approaches the theory of infinite continued fractions a natural question that suggests itself is how to evaluate the expression

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

2002 ◽  
Vol 67 (3) ◽  
pp. 897-909
Author(s):  
David M. Evans ◽  
M. E. Pantano
Keyword(s):  
Growth Rate ◽  
Automorphism Group ◽  
Finite Subset ◽  
Growth Rates ◽  
Algebraic Closure ◽  
Fast Growth ◽  
Permutation Groups ◽  
Natural Numbers ◽  
Categorical Structure ◽  
Image Position

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

On the nth root set of an element in a connected semisimple Lie group

1979 ◽  
Vol 86 (2) ◽  
pp. 219-226 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Lie Group ◽  
Semisimple Lie Group ◽  
Natural Numbers ◽  
Image Position

For any group G and x ∈ G, and any n ∈ ℕ (the natural numbers) leti.e. the set of all nth roots of x in G.

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

Applications of Strict Π11 predicates to infinitary logic

1969 ◽  
Vol 34 (3) ◽  
pp. 409-423 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Normal Form ◽  
Infinitary Logic ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Image Position

Consider the predicate of natural numbers defined by: where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.

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