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-language $L\left(𝒜\right)$L(x1d49c;) accepted by a Büchi 1-counter automaton $𝒜$x1d49c;. We prove the following surprising result: there exists a 1-counter Büchi automaton $𝒜$x1d49c; such that the cardinality of the complement $L{\left(𝒜\right)}^{-}$L(𝒜) −  of the ω-language $L\left(𝒜\right)$L(𝒜) is not determined...

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-language $L\left(𝒜\right)$L(𝒜) accepted by a Büchi 1-counter automaton $𝒜$𝒜. We prove the following surprising result: there exists a 1-counter Büchi automaton $𝒜$𝒜 such that the cardinality of the complement $L{\left(𝒜\right)}^{-}$L(𝒜) −  of the ω-language $L\left(𝒜\right)$L(𝒜) is not determined by ZFC: (1) There is a model V1...