scholarly journals On cardinalities and compact closures

2017 ◽  
Vol 18 (1) ◽  
pp. 107
Author(s):  
Mike Krebs
Keyword(s):  
Set Theory ◽  
Title Page ◽  
General Topology ◽  
Real Numbers ◽  
Hausdorff Topology ◽  
Compact Closure ◽  
Regular Cardinality ◽  
Infinite Set

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>We show that there exists a Hausdorff topology on the set </span><span>R </span><span>of real numbers such that a subset </span><span>A </span><span>of </span><span>R </span><span>has compact closure if and only if </span><span>A </span><span>is countable. More generally, given any set </span><span>X </span><span>and any infinite set </span><span>S</span><span>, we prove that there exists a Hausdorff topology on </span><span>X </span><span>such that a subset </span><span>A </span><span>of </span><span>X </span><span>has compact closure if and only if the cardinality of </span><span>A </span><span>is less than or equal to that of </span><span>S</span><span>. When we attempt to replace “than than or equal to” in the preceding statement with “strictly less than,” the situation is more delicate; we show that the theorem extends to this case when </span><span>S </span><span>has regular cardinality but can fail when it does not. This counterexample shows that not every bornology is a bornology of compact closure. These results lie in the intersection of analysis, general topology, and set theory. </span></p></div></div></div>

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Products of some special compact spaces and restricted forms of AC

2010 ◽  
Vol 75 (3) ◽  
pp. 996-1006 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis
Keyword(s):  
Set Theory ◽  
Closed Subset ◽  
Choice Function ◽  
Ordinal Number ◽  
Axiom Of Choice ◽  
Finite Support ◽  
Compact Spaces ◽  
The Axiom Of Choice ◽  
Infinite Set

AbstractWe establish the following results:1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

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.

sets of reals

1997 ◽  
Vol 62 (4) ◽  
pp. 1379-1428 ◽  
Author(s):  
Joan Bagaria ◽  
W. Hugh Woodin
Keyword(s):  
Set Theory ◽  
Real Numbers ◽  
Borel Sets ◽  
Definable Sets ◽  
Analytic Sets ◽  
Basic Facts ◽  
Continuous Images

Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC.When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the , and using forcing constructions we will produce models of ZFC where, for some n, all , sets have some specified property. For the definition and basic facts about the projective classes , and , as well as the Kleene (or lightface) classes , and , we refer the reader to Moschovakis [19].The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpiński [17] showed that all analytic (i.e., ) sets of reals are Lebesgue measurable and have the property of Baire.

Fuzzy Logic in the Broad Sense

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Human Reasoning ◽  
Human Beings ◽  
Real Numbers ◽  
New Directions ◽  
Gradual Development

The chapter begins by introducing the important and useful distinction between the research agendas of fuzzy logic in the narrow and the broad senses. The chapter deals with the latter agenda, whose ultimate goal is to employ intuitive fuzzy set theory for emulating commonsense human reasoning in natural language and other unique capabilities of human beings. Restricting to standard fuzzy sets, whose membership degrees are real numbers in the unit interval [0,1], the chapter describes how this broad agenda has become increasingly specific via the gradual development of standard fuzzy set theory and the associated fuzzy logic. An overview of currently recognized nonstandard fuzzy sets, which open various new directions in fuzzy logic, is presented in the last section of this chapter.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

Flat sets

1994 ◽  
Vol 59 (3) ◽  
pp. 1012-1021
Author(s):  
Arthur D. Grainger
Keyword(s):  
Set Theory ◽  
Ordinal Number ◽  
Transfer Principle ◽  
Nonstandard Model ◽  
Infinite Set

AbstractLet X be a set, and let be the superstructure of X, where X0 = X and is the power set of X) for n ∈ ω. The set X is called a flat set if and only if for each x ∈ X, and x ∩ ŷ = ø for x, y ∈ X such that x ≠ y. where is the superstructure of y. In this article, it is shown that there exists a bijection of any nonempty set onto a flat set. Also, if is an ultrapower of (generated by any infinite set I and any nonprincipal ultrafilter on I), it is shown that is a nonstandard model of X: i.e., the Transfer Principle holds for and , if X is a flat set. Indeed, it is obvious that is not a nonstandard model of X when X is an infinite ordinal number. The construction of flat sets only requires the ZF axioms of set theory. Therefore, the assumption that X is a set of individuals (i.e., x ≠ ϕ and a ∈ x does not hold for x ∈ X and for any element a) is not needed for to be a nonstandard model of X.

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

Sign in / Sign up

Export Citation Format

Share Document