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.

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.

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

2012 ◽  
Vol 5 (2) ◽  
pp. 269-293 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Paraconsistent Logic ◽  
Large Cardinals ◽  
Axiom Of Choice ◽  
Cardinal Numbers ◽  
Reflection Theorem ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

scholarly journals General-well-ordered sets

1975 ◽  
Vol 19 (1) ◽  
pp. 7-20 ◽  
Author(s):  
J. L. Hickman
Keyword(s):  
Set Theory ◽  
Nonempty Subset ◽  
Minimal Element ◽  
Axiom Of Choice ◽  
Total Order ◽  
Ordered Sets ◽  
Starting Point ◽  
The Axiom Of Choice

It is of course well known that within the framework of any reasonable set theory whose axioms include that of choice, we can characterize well-orderings in two different ways: (1) a total order for which every nonempty subset has a minimal element; (2) a total order in which there are no infinite descending chains.Now the theory of well-ordered sets and their ordinals that is expounded in various texts takes as its definition characterization (1) above; in this paper we commence an investigation into the corresponding theory that takes characterization (2) as its starting point. Naturally if we are to obtain any differences at all, we must exclude the axiom of choice from our set theory. Thus we state right at the outset that we are working in Zermelo-Fraenkel set theory without choice.

A Logarithmic Property for Exponents of Partially Ordered Sets

1978 ◽  
Vol 30 (4) ◽  
pp. 797-807 ◽  
Author(s):  
Dwight Duffus ◽  
Ivan Rival
Keyword(s):  
Partially Ordered Sets ◽  
Independent Interest ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Ordinal Numbers

In an effort to unify the arithmetic of cardinal and ordinal numbers, Garrett Birkhoff [2; 3; 4; 5] (cf. [6]) defined several operations on partially ordered sets of which at least one, (cardinal) exponentiation, is of considerable independent interest: for partially ordered sets P and Q let PQ denote the set of all order-preserving maps of Q to P partially ordered by f ≦ g if and only if f(x) ≦ g(x) for each x ∈ Q.

A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems

1961 ◽  
Vol 13 ◽  
pp. 78-82 ◽  
Author(s):  
Smbat Abian ◽  
Arthur B. Brown
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Prove Theorem ◽  
The Fixed Point Theorem ◽  
The Axiom Of Choice

In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries.Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}.Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.

Edwin W. Miller. On a property of families of sets. English with Polish summary. Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego (Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie), Class III, vol. 30 (1937), pp. 31–38. - Ben Dushnik and Miller E. W.. Partially ordered sets. American journal of mathematics, vol. 63 (1941), pp. 600–610. - P. Erdős. Some set-theoretical properties of graphs. Revista, Universidad Nacional de Tucumán, Serie A, Matemáticas y física teórica, vol. 3 (1942), pp. 363–367. - G. Fodor. Proof of a conjecture of P. Erdős. Acta scientiarum mathematicarum, vol. 14 no. 4 (1952), pp. 219–227. - P. Erdős and Rado R.. A partition calculus in set theory. Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–489. - P. Erdős and Rado R.. Intersection theorems for systems of sets. The journal of the London Mathematical Society, vol. 35 (1960), pp. 85–90. - A. Hajnal. Some results and problems on set theory. Acta mathematica Academiae Scientiarum Hungaricae, vol. 11 (1960), pp. 277–298. - P. Erdős and Hajnal A.. On a property of families of sets. Acta mathematica Academiae Scientiarum Hungaricae, vol. 12 (1961), pp. 87–123. - A. Hajnal. Proof of a conjecture of S. Ruziewicz. Fundamenta mathematicae, vol. 50 (1961), pp. 123–128. - P. Erdős, Hajnal A. and Rado R.. Partition relations for cardinal numbers. Acta mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196. - P. Erdős and Hajnal A.. On a problem of B. Jónsson. Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, vol. 14 (1966), pp. 19–23. - P. Erdős and Hajnal A.. On chromatic number of graphs and set-systems. Acta mathematica Academiae Scientiarum Hungaricae, vol. 17 (1966), pp. 61–99.

1995 ◽  
Vol 60 (2) ◽  
pp. 698-701
Author(s):  
James E. Baumgartner
Keyword(s):  
Set Theory ◽  
American Mathematical Society ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Class Iii ◽  
Ordered Sets ◽  
Partition Calculus ◽  
Cardinal Numbers ◽  
Set Systems ◽  
Partition Relations

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

2010 ◽  
Vol 3 (1) ◽  
pp. 71-92 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Paraconsistent Logic ◽  
Peano Arithmetic ◽  
Large Cardinals ◽  
Cardinal Numbers ◽  
Naive Set Theory ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
The Continuum

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Structure Results for Function Lattices

1978 ◽  
Vol 30 (02) ◽  
pp. 392-400 ◽  
Author(s):  
D. Duffus ◽  
B. Jónsson ◽  
I. Rival
Keyword(s):  
Direct Product ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Cardinal Numbers

For partially ordered sets X and F let Yx denote the set of all order-preserving maps of X to Y partially ordered by f ≦ g if and only if f(x)≦ g (x) for each x ∈ X [1; 4; 6]. If X is unordered then Yx is the usual direct product of partially ordered sets, while if both X and Y are finite unordered sets then Yx is the commonplace exponent of cardinal numbers. This generalized exponentiation has an important vindication especially for those partially ordered sets that are lattices.

Real functions on the family of all well-ordered subsets of a partially ordered set

1983 ◽  
Vol 48 (1) ◽  
pp. 91-96 ◽  
Author(s):  
Stevo Todorčević
Keyword(s):  
Set Theory ◽  
Real Function ◽  
Partially Ordered Set ◽  
Initial Segment ◽  
Ordered Set ◽  
Monotone Mapping ◽  
Ordered Sets ◽  
Partially Ordered ◽  
The Family ◽  
Aronszajn Trees

Definition 1 (Kurepa [3, p. 99]). Let E be a partially ordered set. Then σE denotes the set of all bounded well-ordered subsets of E. We consider σE as a partially ordered set with ordering defined as follows: st if and only if s is an initial segment of t.Then σE is a tree, i.e., {s ∈ σ E∣ st} is well-ordered for every t ∈ σE. The trees of the form αE were extensively studied by Kurepa in [3]–[10]. For example, in [4], he used σQ and σR to construct various sorts of Aronszajn trees. (Here Q and R denote the rationals and reals, respectively.) While considering monotone mapping between some kind of ordered sets, he came to the following two questions several times:P.1. Does there exist a strictly increasing rational function on σQ? (See [4, Problème 2], [5, p. 1033], [6, p. 841], [7, Problem 23.3.3].)P.2. Let T be a tree in which every chain is countable and every level has cardinality <2ℵ0. Does there exist a strictly increasing real function on T? (See [6, p. 246] and [7].)It is known today that Problem 2 is independent of the usual axioms of set theory (see [1]). Concerning Problem 1 we have the following.

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