1. The axiom of choice. The object of this paper is to consider the possibUity of setting up a logic in which the axiom of choice is false. The way of approach is through the second ordinal class, in connection with which there appear certain alternatives to the axiom of choice. But these alternatives have consequences not only with regard to the second ordinal class but also with regard to oth...

Suppose stfis a collection of subsets of the unit interval and, for A e s/, \iA is a Borel measure on A which vanishes on points and gives A measure 1. The system pA (A e jtf) is called a coherent system if pA(C) = nA(B)pB(C) whenever A dSdC are in .Wand all terms are defined. The existence of a coherent system for the collection of perfect sets is shown to be independent of Zermelo-Fraenkel se...

When using the Curry-Howard correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem : to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem has been solved for the axiom of excluded midd...

A geometric approach is introduced to explain phenomena that can arise with Luce’s choice axiom; e.g., differences occur when determining the likelihood of a ranking by starting with the “best-first,” or “worst-first” alternative. As shown, the problem is caused by the way we compute pairwise probabilities: it forces “bestfirst” and “worst-first” computations to use different information from a...

There are many topological results in Zermelo-Fraenkel set theory including the axiom of choice (ZFC) that are not true in the absence of choice, i. e. in ZF. Even if we restrict our attention to many “familiar” topological results are not provable in ZF, although in most cases their validity follows from a weaker version of the axiom of choice, CC( ). Definition 0.1 The axiom of countable choi...