On the fixed points of a computable function having no computable fixed points

Abstract:

The Brouwer fixed-point theorem is essentially non-constructive since the counter-examples of Orevkov and Baigger imply that there is no general procedure for finding the fixed point – by constructing a computable function which does not fix any (Turing) computable point in the unit square. A new computable enumeration property is proposed which guarantees that whenever a computable function has only non-computable fixed points, a set with this property always contains either none or infinitely many of the fixed points. The counter-examples of Orevkov and Baigger as well as the relation of the problem with the König lemma will be briefly discussed as well.