We shall start with some definitions from topology. First of all, a metric space is a topological space whose topology is determined by a metric. A metric on a topological space X is a function d from X × X to R , the reals, which has the following properties: For all x, y, z ∈ X , (a) d(x, y) ≥ 0, (b) d(x, x) = 0, (c) if d(x, y) = 0, then x = y, (d) d(x, y) = d(y, x), and (e) d(x, y) + d(y, z)...

A b s t r a c t. We show in set-theory ZF that the axiom of choice is equivalent to the statement every bipartite connected graph has a spanning sub-graph omitting some complete finite bipartite graph K n,n , and in particular it is equivalent to the fact that every connected graph has a spanning cycle-free graph (possibly non connected). We consider simple undirected loop-free graphs. A forest...

This theorem is well known to be equivalent to the axiom of choice (though there does not seem to be a proof of this fact in the literature) and it has been suggested as an alternative for this axiom. The purpose of this note (which is purely methodological) is to propose a simpler but equivalent formulation of (A) as a substitute for the Zermelo axiom. The simplicity lies in the fact that we m...

We study the connection between the axiom of choice and the principles of existence of enough projective and injective abehan groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other th...

In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial.

We show that for all natural numbers n, the theory “ZF +DCאn + אω is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + אω1 is an ω2-Rowbottom cardinal carrying an ω2-Rowbottom filter and ω1 is regular” has the same consistency strength as the theory “ZFC + There...

function on A is thus a choice of an element of the variable set A at each stage; in other words, a choice function on A is just a variable (or global) element of A . The axiom of choice (AC) asserts that if each member of a family A is nonempty, then there is a choice function on A . Metaphorically, then, the axiom of choice asserts that any family of sets with an element at each stage has a v...