  We prove the following surprising result: there exist a 1-counter B\"uchi
automaton and a 2-tape B\"uchi automaton such that the \omega-language of the
first and the infinitary rational relation of the second in one model of ZFC
are \pi_2^0-sets, while in a different model of ZFC both are analytic but non
Borel sets.
  This shows that the topological complexity of an \omega-language accepted by
a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by
a 2-tape B\"uchi 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 B\"uchi 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.
