Hintikka’s Take on the Axiom of Choice and the Constructivist Challenge

Abstract

In the present paper we confront Martin- Löf’s analysis of the axiom of choice with J. Hintikka’s standing on this axiom. Hintikka claims that his game theoretical semantics (GTS) for Independence Friendly Logic (IF logic) justifies Zermelo’s axiom of choice in a first-order way perfectly acceptable for the constructivists. In fact, Martin- Löf’s results lead to the following considerations:

1. Hintikka preferred version of the axiom of choice is indeed acceptable for the constructivists and its meaning does not involve higher order logic.
2. However, the version acceptable for constructivists is based on an intensional take on functions. Extensionality is the heart of the classical understanding of Zermelo’s axiom and this is the real reason behind the constructivist rejection of it.
3. More generally, dependence and independence features that motivate IF-Logic, can be formulated within the frame of constructive type theory (CTT) without paying the price of a system that is neither axiomatizable nor has an underlying theory of inference – logic is about inference after all.

We conclude pointing out that recent developments in dialogical logic show that the CTT approach to meaning in general and to the axiom of choice in particular is very natural to game theoretical approaches where (standard) metalogical features are explicitly displayed at the object language-level. Thus, in some way, this vindicates, albeit in quite of a different manner, Hintikka’s plea for the fruitfulness of game-theoretical semantics in the context of the foundations of mathematics.