Cantor’s theorem

Cantor’s theorem states that the cardinal number (‘size’) of the set of subsets of any set is greater than the cardinal number of the set itself. So once the existence of one infinite set has been proved, sets of ever increasing infinite cardinality can be generated. The philosophical interest of this result lies (1) in the foundational role it played in Cantor’s work, prior to the axiomatization of set theory, (2) in the similarity between its proof and arguments which lead to the set-theoretic paradoxes, and (3) in controversy between intuitionist and classical mathematicians concerning what exactly its proof proves.

Products of some special compact spaces and restricted forms of AC

10.2178/jsl/1278682212 ◽

2010 ◽

Vol 75 (3) ◽

pp. 996-1006 ◽

Cited By ~ 2

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 ◽

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.

A Note on z-Ideals and z°-Ideals of the Formal Power Series Rings and Polynomial Rings in an Infinite Set of Indeterminates

Algebra Colloquium ◽

10.1142/s1005386720000413 ◽

2020 ◽

Vol 27 (03) ◽

pp. 495-508

Author(s):

Ahmed Maatallah ◽

Ali Benhissi

Keyword(s):

Power Series ◽

Formal Power Series ◽

Cardinal Number ◽

Prime Ideals ◽

Infinite Cardinal ◽

Series Ring ◽

Formal Power Series Ring ◽

Power Series Rings ◽

Formal Power ◽

Let A be a ring. In this paper we generalize some results introduced by Aliabad and Mohamadian. We give a relation between the z-ideals of A and those of the formal power series rings in an infinite set of indeterminates over A. Consider A[[XΛ]]3 and its subrings A[[XΛ]]1, A[[XΛ]]2, and A[[XΛ]]α, where α is an infinite cardinal number. In fact, a z-ideal of the rings defined above is of the form I + (XΛ)i, where i = 1, 2, 3 or an infinite cardinal number and I is a z-ideal of A. In addition, we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients. As a natural result, we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.

Flat sets

10.2307/2275925 ◽

1994 ◽

Vol 59 (3) ◽

pp. 1012-1021

Author(s):

Arthur D. Grainger

Keyword(s):

Set Theory ◽

Ordinal Number ◽

Transfer Principle ◽

Nonstandard Model ◽

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.

Clubs: A Student-Presented Mathematics Club Program—Infinities of Numbers

Mathematics Teacher ◽

10.5951/mt.74.9.0711 ◽

1981 ◽

Vol 74 (9) ◽

pp. 711-712

Author(s):

Paul M. Nemecek

Keyword(s):

Cardinal Number ◽

Mathematical Discussion ◽

One To One ◽

There is widespread use of the words infinite or infinitely many by students; yet there seldom occurs a sound mathematical discussion of the topic. The proof of the existence of infinity, accomplished in the last hundred years, allows an important opportunity to discuss the concepts of number, cardinal number, infinite set, and one-to-one correspondence.

A congruence-free semigroup associated with an infinite cardinal number

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050001595x ◽

1983 ◽

Vol 93 (3-4) ◽

pp. 245-257 ◽

Cited By ~ 7

Author(s):

M. Paula O. Marques

Keyword(s):

Homomorphic Image ◽

Free Semigroup ◽

Cardinal Number ◽

Infinite Cardinal ◽

Rees Factor ◽

Infinite Cardinality ◽

The Ideal

SynopsisLet X be a set with infinite cardinality m and let Qm be the semigroupof balanced elements in ℐ(X), as described by Howie. If I is the ideal{αεQm:|Xα|<m} then the Rees factor Pm = Qm/I is O-bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is congruence-free. Moreover, has depth 4, in the sense that [E()]4 = , [E()]3≠

On a cardinal equation in set theory

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700044713 ◽

1972 ◽

Vol 6 (3) ◽

pp. 447-457 ◽

Cited By ~ 1

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).

Omitting types in set theory and arithmetic

10.1017/s0022481200051719 ◽

1976 ◽

Vol 41 (1) ◽

pp. 25-32 ◽

Cited By ~ 1

Author(s):

Julia F. Knight

Keyword(s):

Set Theory ◽

Conservative Extension ◽

Alternate Proof ◽

Hanf Number ◽

Omitting Types ◽

Infinite Set ◽

Models Of Set Theory

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

Consequences of arithmetic for set theory

10.2307/2275247 ◽

1994 ◽

Vol 59 (1) ◽

pp. 30-40 ◽

Cited By ~ 13

Author(s):

Lorenz Halbeisen ◽

Saharon Shelah

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Power Set ◽

The Axiom Of Choice ◽

AbstractIn this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals and , either ≤ or ≤ . However, in ZF this is no longer so. For a given infinite set A consider seq1-1(A), the set of all sequences of A without repetition. We compare |seq1-1(A)|, the cardinality of this set, to ||, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF ⊢ ∀A(| seq1-1(A)| ≠ ||), and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then | fin(B)| < | (B*)| even though the existence for some infinite set B* of a function ƒ from fin(B*) onto (B*) is consistent with ZF.

Normal subgroups of infinite symmetric groups, with an application to stratified set theory

10.2178/jsl/1231082300 ◽

2009 ◽

Vol 74 (1) ◽

pp. 17-26 ◽

Cited By ~ 1

Author(s):

Nathan Bowler ◽

Thomas Forster

Keyword(s):

Set Theory ◽

Symmetric Groups ◽

Axiom Of Choice ◽

Normal Subgroups ◽

Standard Analysis ◽

Small Index ◽

The Axiom Of Choice ◽

Two Stages ◽

Infinite Set ◽

Stratified Set

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

Grothendieck Universes

Formalized Mathematics ◽

10.2478/forma-2020-0018 ◽

2020 ◽

Vol 28 (2) ◽

pp. 211-215

Author(s):

Karol Pąk

Keyword(s):

Set Theory ◽

Proper Subset ◽

Axiom A ◽

First Order ◽

Transitive Set ◽

Mizar Mathematical Library ◽

The Relationship ◽

Summary The foundation of the Mizar Mathematical Library [2], is first-order Tarski-Grothendieck set theory. However, the foundation explicitly refers only to Tarski’s Axiom A, which states that for every set X there is a Tarski universe U such that X ∈ U. In this article, we prove, using the Mizar [3] formalism, that the Grothendieck name is justified. We show the relationship between Tarski and Grothendieck universe. First we prove in Theorem (17) that every Grothendieck universe satisfies Tarski’s Axiom A. Then in Theorem (18) we prove that every Grothendieck universe that contains a given set X, even the least (with respect to inclusion) denoted by GrothendieckUniverseX, has as a subset the least (with respect to inclusion) Tarski universe that contains X, denoted by the Tarski-ClassX. Since Tarski universes, as opposed to Grothendieck universes [5], might not be transitive (called epsilon-transitive in the Mizar Mathematical Library [1]) we focused our attention to demonstrate that Tarski-Class X ⊊ GrothendieckUniverse X for some X. Then we show in Theorem (19) that Tarski-ClassX where X is the singleton of any infinite set is a proper subset of GrothendieckUniverseX. Finally we show that Tarski-Class X = GrothendieckUniverse X holds under the assumption that X is a transitive set. The formalisation is an extension of the formalisation used in [4].