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 the statement CCFC = “the character of a maximal free filter $F$ of closed sets in a $T_1$ space $(X,T)$ is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE${}_0$ = “closed filters in a $T_0$ space $(X,T)$ extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter $ℱ$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base”, “for every set $A\ne \varnothing$,...

Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that $(V_{\alpha },\in )$ satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a) LM. (b)...

We show: (i) The countable axiom of choice $\mathrm{𝐂𝐀𝐂}$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $\mathrm{𝐂𝐌𝐂}$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $\mathrm{𝐀𝐂}$ is equivalent to each one of the...

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 ${\aleph }_{\alpha }$ or $2^{{\aleph }_{\alpha }}$; • finite sums...

A theorem is proved which could be considered as a bridge between the combinatorics which have a beginning in the dyadic spaces theory and the partition calculus.

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...

We investigate the question whether a system ${\left(E_i\right)}_{i\in I}$ of homogeneous linear equations over $\mathbb{Z}$ is non-trivially solvable in $\mathbb{Z}$ provided that each subsystem ${\left(E_j\right)}_{j\in J}$ with $\left|J\right|\le c$ is non-trivially solvable in $\mathbb{Z}$ where $c$ is a fixed cardinal number such that $c<\left|I\right|$. 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., $\left|I\right|={\aleph }_0$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\le {\aleph }_0$ is ‘No relatively consistent...

We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.

Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain $T_{\alpha }:\alpha <\lambda$ of infinite subsets of ω, there exists $ℳ\subset {\left[\omega \right]}^{\omega }$, an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain $T_{\alpha }:\alpha <\lambda$, hence a ψ-space with Stone-Čech remainder...

Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector...

We study several perfect set properties of the Baire space which follow from the Ramsey property $\omega \to {\left(\omega \right)}^{\omega }$. In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.