A Modal Theory of Free Choice Sequences

Ethan Brauer

Free choice sequences (also called ‘infinitely  proceeding sequences’) are a concept from intuitionistic mathematics that are central to the development of the intuitionistic theory of the continuum. Free choice sequences have, however, been largely rejected or ignored both by classical and other constructive mathematicians. In this talk I argue that the objections to free choice sequences can be overcome by grounding the concept in our temporal intuition and formalizing the theory in modal logic. I will present a theory of free choice sequences as a modal extension of classical second-order arithmetic. The resulting theory is able to prove modal versions of the intuitionist’s axioms for so-called lawless sequences, suffices for the development of a theory of real number generators, and captures many results distinctive of intuitionistic analysis including the non-existence of functions on real numbers with definable discontinuities.