Independence, randomness and the axiom of choice

Independence, Randomness and the Axiom of Choice

The Axiom of Choice

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...

The Axiom of Choice

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...

The Congruence Axiom and Path Independence

This paper examines the connection between path independence and transitive rationalization of a choice function in a general domain where a choice function may not include all possible non-empty subsets of the universal set. Describing a sequential choice procedure, this paper shows that the requirement of path independence of a choice function is equivalent to Richter's Congruence Axiom and t...

The independence of Tarski's Euclidean axiom

Tarski’s axioms of plane geometry are formalized and, using the standard real Cartesian model, shown to be consistent. A substantial theory of the projective plane is developed. Building on this theory, the Klein–Beltrami model of the hyperbolic plane is defined and shown to satisfy all of Tarski’s axioms except his Euclidean axiom; thus Tarski’s Euclidean axiom is shown to be independent of hi...