  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
omega-language $L(A)$ accepted by a B\"uchi 1-counter automaton $A$. We prove
the following surprising result: there exists a 1-counter B\"uchi automaton $A$
such that the cardinality of the complement $L(A)^-$ of the omega-language
$L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which
$L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A)^-$ has
cardinal $2^{\aleph_0}$. (3). There is a model $V_3$ of ZFC in which $L(A)^-$
has cardinal $\aleph_1$ with $\aleph_0<\aleph_1<2^{\aleph_0}$. We prove a very
similar result for the complement of an infinitary rational relation accepted
by a 2-tape B\"uchi automaton $B$. As a corollary, this proves that the
Continuum Hypothesis may be not satisfied for complements of 1-counter
omega-languages and for complements of infinitary rational relations accepted
by 2-tape B\"uchi automata. We infer from the proof of the above results that
basic decision problems about 1-counter omega-languages 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
omega-language (respectively, infinitary rational relation) is countable is in
$\Sigma_3^1 \setminus (\Pi_2^1 \cup \Sigma_2^1)$. This is rather surprising if
compared to the fact that it is decidable whether an infinitary rational
relation is countable (respectively, uncountable).
