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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 () does not imply “the Tychonoff product , 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...
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...
We show that it is consistent with ZF that there is a dense-in-itself compact metric space which has the countable chain condition (ccc), but is neither separable nor second countable. It is also shown that has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply .
In set theory without the axiom of choice (), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC (AC for linearly ordered families of nonempty sets)—and hence AC (AC for well-ordered families of nonempty sets)— (where is an uncountable regular cardinal), and “for every infinite set , there is a bijection ”, implies the statement “there exists a field such that every vector...
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 , 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...
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.
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 or ; • finite sums...
We investigate the question whether a system of homogeneous linear equations over is non-trivially solvable in provided that each subsystem with is non-trivially solvable in where is a fixed cardinal number such that . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., being finite). (b) The answer is ‘No’ in the denumerable case (i.e., and a natural number). (c) The answer in case that is uncountable and is ‘No relatively consistent...
We show that the statement CCFC = “” is equivalent to the CMC and, the axiom of choice AC is equivalent to the statement CFE = “”. We also show that AC is equivalent to each of the assertions: “”, “” and “”.
In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product , 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.
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " is countably compact" and " is compact"
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...
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.
We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.
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 has a ternary partition (see Section 1, Definition 2) then the Russell cardinal 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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