Hereditarily finite Finsler sets

1990 ◽  
Vol 55 (2) ◽  
pp. 700-706
Author(s):  
David Booth
Keyword(s):  
Set Theory ◽  
Finite Collection ◽  
Present Stage ◽  
Finite Sets ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Mathematical Operations ◽  
Exhaustive List

Hereditarily finite sets are sets which are finite, whose members are finite, the members of whose members are finite, and so on. In ZF there are but countably many such sets; they constitute Vω. Were ZF to lose its axiom of regularity, however, one could not guarantee that the number of hereditarily finite sets would remain countable.In Mostowski set theory, in which atomic sets are permissible, each atom, in isolation, would form a hereditarily finite collection. The number of hereditarily finite sets could be anything one should choose.Even in a world that did not permit the free adjunction of arbitrary, meaningless atoms, the number of hereditarily finite sets could remain large. In Finsler set theory, it is shown as Theorem 22, below, that there are uncountably many hereditarily finite sets.The reader who is interested in this paradoxical sounding fact can turn directly to §4 after grasping these introductory concepts. §3 is an exhaustive list of the smallest Finsler sets; it is hoped that this list will prove useful in checking future attempts to classify the finite Finsler sets.Finsler set theory is not a firmly axiomatized theory. It is, at its present stage, a family of theories undergoing evolution. It permits the usual mathematical operations with sets. One can employ ordinal numbers, cardinal numbers, and the usual methods of Cantorian set theory freely. But there is a somewhat different interpretation attached to the concept “set” than one is used to in Zermelo-Fraenkel set theory, ZF.

Type-raising operations on cardinal and ordinal numbers in Quine's “New foundations”

1973 ◽  
Vol 38 (1) ◽  
pp. 59-68 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Set Theory ◽  
Initial Segment ◽  
Initial Section ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Basic Facts ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Increasing Function ◽  
Standard Operations

In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, T(n) on cardinal numbers and U(α) on ordinal numbers, which are defined by the equationsfor each set x and well ordering R. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas T(x) = y and U(x) = y are stratified when y is assigned a type one higher than x. The importance of T and U stems from the following facts: (i) each of T and U is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand U commute with most of the standard operations on cardinal and ordinal numbers.These basic facts are discussed in §1. In §2 we prove in NF that the exponential function 2n is not 1-1. Indeed, there exist cardinal numbers m and n which satisfyIn §3 we prove the following technical result, which has many important applications. Suppose f is an increasing function from an initial segment S of the set NO of ordinal numbers into NO and that f commutes with U.

Set Theory

2021 ◽  
pp. 25-46
Author(s):  
Adel N. Boules
Keyword(s):  
Vector Space ◽  
Set Theory ◽  
Hilbert Spaces ◽  
Ordered Sets ◽  
Proper Understanding ◽  
Partially Ordered ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic

The chapter is a concise, practical presentation of the basics of set theory. The topics include set equivalence, countability, partially ordered, linearly ordered, and well-ordered sets, the axiom of choice, and Zorn’s lemma, as well as cardinal numbers and cardinal arithmetic. The first two sections are essential for a proper understanding of the rest of the book. In particular, a thorough understanding of countability and Zorn’s lemma are indispensable. Parts of the section on cardinal numbers may be included, but only an intuitive understanding of cardinal numbers is sufficient to follow such topics as the discussion on the existence of a vector space of arbitrary (infinite) dimension, and the existence of inseparable Hilbert spaces. Cardinal arithmetic can be omitted since its applications later in the book are limited. Ordinal numbers have been carefully avoided.

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.

Cardinal Numbers and Finite Sets

Set Theory ◽  
2001 ◽  
pp. 29-43
Author(s):  
Robert L. Vaught
Keyword(s):  
Finite Sets ◽  
Cardinal Numbers

Truth In V for ∃*∀∀-Sentences is Decidable

2006 ◽  
Vol 71 (4) ◽  
pp. 1200-1222 ◽  
Author(s):  
D. Bellé ◽  
F. Parlamento
Keyword(s):  
Set Theory ◽  
Finite Sets ◽  
First Order ◽  
Free Variables ◽  
Order Language

AbstractLet V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and. following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in ∀ but not by finite sets ([5]). and therefore of ∃∃∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for ∃*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows.

scholarly journals On a cardinal equation in set theory

1972 ◽  
Vol 6 (3) ◽  
pp. 447-457 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Linear Ordering ◽  
Finite Sets ◽  
Intrinsic Interest ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Finiteness Criterion

We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).

A formalization of the theory of ordinal numbers

1965 ◽  
Vol 30 (3) ◽  
pp. 295-317 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Extension ◽  
Mathematical Induction ◽  
Predicate Calculus ◽  
Transfinite Induction ◽  
First Order ◽  
Recursive Functions ◽  
Ordinal Numbers ◽  
Primitive Recursive ◽  
Order Predicate Calculus

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.

Theoretical views on Szilard Chalmers reactions in solid systems. II. Parent reformation by billiard-ball collisions

1968 ◽  
Vol 46 (20) ◽  
pp. 3201-3209 ◽  
Author(s):  
W. H. Wong ◽  
D. R. Wiles
Keyword(s):  
Experimental Data ◽  
Thermal Decomposition ◽  
Initial Energy ◽  
Projectile Energy ◽  
Metal Carbonyls ◽  
Present Stage ◽  
Complex Ions ◽  
Billiard Ball ◽  
Lower Limits ◽  
Mathematical Operations

The problem of billiard-ball replacement reactions of atoms centrally located in the molecule has been approached, using the approximation of simple two-body collisions. The reentry process has been separated into steps which can be handled by straightforward mathematical operations. Collision diameters for given energy transfer were calculated using an exponentially screened potential.Species treated include dicyclopentadienyl metals, arene metal carbonyls, and hexacoordinated complex ions, although the method is applicable to many other types of compound. An important result is that the probability of successful billiard-ball replacement is not sensitive to the initial energy, as long as this is not too low. It is concluded that this method is, at its present stage, most useful in calculating lower limits for billiard-ball reformation by following the projectile energy down to ca. 100 eV. Below this energy it is considered that thermal decomposition of the reformed molecule is likely. Results of the calculation are compared with experimental data, and further experiments are suggested by which the contribution of billiard-ball collisions may be directly assessed.

scholarly journals Trudności w opanowaniu odmiany liczebników w nauczaniu języka polskiego jako obcego na przykładzie studentów ukraińsko i rosyjskojęzycznych.

2021 ◽  
pp. 23-36
Author(s):  
Ольга Попова
Keyword(s):  
Comparative Method ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Causes Of Errors ◽  
Syntactic Relations

The aim of this paper is to describe the most common errors made by Ukrainian- and Russian-speaking people while learning infl ection of numerals for person and their causes. The applied comparative method permits a closer look at formal and syntactic relations as well as collocations of numerals and specifi c lexeme classes (this regards primarily agreement between cardinal numbers and masculine personal verbs, compound structures with the numeral jeden (one) and numerals showing adjectival infl ection). This study concentrates on a comparison of cardinal and ordinal numbers in Ukrainian, Russian and Polish, which allowed an analysis of the causes of errors made while learning the infl ection of numerals by Ukrainian- and Russian-speaking students.

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