Computable counter-examples to the Brouwer fixed-point theorem

Abstract

This paper is an overview of results that show the Brouwer fixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov and Baigger, imply that there is no procedure for finding the fixed point in general by giving an example of a computable function which does not fix any computable point. Research in reverse mathematics has shown the BFPT to be equivalent to the weak König lemma in RCA${}_0$ (the system of recursive comprehension) and this result is illustrated by relating the weak König lemma directly to the Baigger example.

10 pages; to appear in local proceedings of Computability in Europe 2008: Logic and Theory of Algorithms