An argument often given for adopting the Axiom of Choice as an axiom is that it has a lot of obviously true consequences. This looks like a legitimate application of the practice of Inference to the Best Explanation. However, the standard examples of obvious-truths-following-from-AC all turn out, on closer inspection, to involve a fallacy of equivocation.

We charaterize the choice correspondences that can be rationalized by a procedure that is a refinement of the prudent choices exposed in [Houy, 2008]. Our characterization is made by means of the usual expansion axiom and by a weakening of the usual contraction axiom α.

We develop a technology for investigation of natural forcing extensions of the model L(R) which satisfy such statements as “there is an ultrafilter” or “there is a total selector for the Vitali equivalence relation”. The technology reduces many questions about ZF implications between consequences of the axiom of choice to natural ZFC forcing problems.

We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice Our interpretation seems computationally more direct than the one based on G odel s Dialectica interpretation Interestingly this interpretation uses a re nement of the realizibility semantics of the absurdity proposition which is not interpreted as the empty type here We also s...

We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.

This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

In this paper we introduce a new axiom scheme, the Relation Reflection Scheme (RRS), for constructive set theory. Constructive set theory is an extensional set theoretical setting for constructive mathematics. A formal system for constructive set theory was first introduced by Myhill in [8]. In [1, 2, 3] I introduced a formal system CZF that is closely related to Myhill’s formal system and gave...

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.