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On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product 2 ℝ

Eleftherios Tachtsis — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ( A C W O ) does not imply “the Tychonoff product 2 ℝ , where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets of reals imply...

On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF

Eleftherios Tachtsis — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

In ZF (i.e. Zermelo-Fraenkel set theory without the Axiom of Choice AC), we investigate the relationship between UF(ω) (there exists a free ultrafilter on ω) and the statements "there exists a free ultrafilter on every Russell-set" and "there exists a Russell-set A and a free ultrafilter ℱ on A". We establish the following results: 1. UF(ω) implies that there exists a free ultrafilter on every Russell-set. The implication is not reversible in ZF. 2. The statement...

On certain non-constructive properties of infinite-dimensional vector spaces

Eleftherios Tachtsis — 2018

Commentationes Mathematicae Universitatis Carolinae

In set theory without the axiom of choice ( AC ), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC LO (AC for linearly ordered families of nonempty sets)—and hence AC WO (AC for well-ordered families of nonempty sets)— DC ( < κ ) (where κ is an uncountable regular cardinal), and “for every infinite set X , there is a bijection f : X → { 0 , 1 } × X ”, implies the statement “there exists a field F such that every vector...

Disasters in metric topology without choice

Eleftherios Tachtsis — 2002

Commentationes Mathematicae Universitatis Carolinae

We show that it is consistent with ZF that there is a dense-in-itself compact metric space ( X , d ) which has the countable chain condition (ccc), but X is neither separable nor second countable. It is also shown that X has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply .

On the set-theoretic strength of the n-compactness of generalized Cantor cubes

Paul Howard;  Eleftherios Tachtsis — 2016

Fundamenta Mathematicae

We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product 2 X , where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set...

On rigid relation principles in set theory without the axiom of choice

Paul Howard;  Eleftherios Tachtsis — 2016

Fundamenta Mathematicae

We study the deductive strength of the following statements: 𝖱𝖱: every set has a rigid binary relation, 𝖧𝖱𝖱: every set has a hereditarily rigid binary relation, 𝖲𝖱𝖱: every set has a strongly rigid binary relation, in set theory without the Axiom of Choice. 𝖱𝖱 was recently formulated by J. D. Hamkins and J. Palumbo, and 𝖲𝖱𝖱 is a classical (non-trivial) 𝖹𝖥𝖢-result by P. Vopěnka, A. Pultr and Z. Hedrlín.

On the solvability of systems of linear equations over the ring ℤ of integers

Horst Herrlich;  Eleftherios Tachtsis — 2017

Commentationes Mathematicae Universitatis Carolinae

We investigate the question whether a system ( E i ) i ∈ I of homogeneous linear equations over ℤ is non-trivially solvable in ℤ provided that each subsystem ( E j ) j ∈ J with | J | ≤ c is non-trivially solvable in ℤ where c is a fixed cardinal number such that c < | I | . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., I being finite). (b) The answer is ‘No’ in the denumerable case (i.e., | I | = ℵ 0 and c a natural number). (c) The answer in case that I is uncountable and c ≤ ℵ 0 is ‘No relatively consistent...

On the number of Russell’s socks or 2 + 2 + 2 + ⋯ = ?

Horst Herrlich;  Eleftherios Tachtsis — 2006

Commentationes Mathematicae Universitatis Carolinae

The following question is analyzed under the assumption that the Axiom of Choice fails badly: Given a countable number of pairs of socks, then how many socks are there? Surprisingly this number is not uniquely determined by the above information, thus giving rise to the concept of Russell-cardinals. It will be shown that: • some Russell-cardinals are even, but others fail to be so; • no Russell-cardinal is odd; • no Russell-cardinal is comparable with any cardinal of the form ℵ α or 2 ℵ α ; • finite sums...

Remarks on the Stone Spaces of the Integers and the Reals without AC

Horst Herrlich;  Kyriakos Keremedis;  Eleftherios Tachtsis — 2011

Bulletin of the Polish Academy of Sciences. Mathematics

In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2 ( X ) , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

Countable Compact Scattered T₂ Spaces and Weak Forms of AC

Kyriakos Keremedis;  Evangelos Felouzis;  Eleftherios Tachtsis — 2006

Bulletin of the Polish Academy of Sciences. Mathematics

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

Horst Herrlich;  Paul Howard;  Eleftherios Tachtsis — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice ( AC ), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y → X.

On special partitions of Dedekind- and Russell-sets

Horst Herrlich;  Paul Howard;  Eleftherios Tachtsis — 2012

Commentationes Mathematicae Universitatis Carolinae

A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal a has a ternary partition (see Section 1, Definition 2) then the Russell cardinal a + 2 fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell...

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