The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter Büchi automaton and a 2-tape Büchi automaton such that the ω-language of the first and the infinitary rational relation of the second in one model of ZFC are Π2-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an ω-language accepted by a 1-counter Büchi automaton or of an infinitary rational relation accepted by a 2-tape Büchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by Büchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author.