Some problems in automata theory which depend on the models of set theory

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-languageL(A) accepted by a Büchi 1-counter automaton A. We prove the following surprising result: there exists a 1-counter Büchi automaton A such that the cardinality of the complementL(A) of the ω-languageL(A) is not determined by ZFC: (1). There is a model V1 of ZFC in which L(A) is countable. (2). There is a model V2 of ZFC in which L(A) has cardinal 20 . (3). There is a model V3 of ZFC in which L(A) has cardinal א1 with א0 < א1 < 20 . We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton B. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter ωlanguages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ωlanguage (respectively, infinitary rational relation) is countable is in Σ 3 \ (Π 2 ∪ Σ 2 ). This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).