We prove that there exist some 1-counter Büchi automata A_n for which some elementary properties are independent of theories like T_n =: ZFC + " There exist (at least) n inaccessible cardinals " , for integers n ≥ 1. In particular , if T_n is consistent, then " L(A_n) is Borel " , " L(A_n) is arithmetical " , " L(A_n) is ω-regular " , " L(A_n) is deterministic " , and " L(A_n) is unambiguous " are provable from ZFC + " There exist (at least) n+1 inaccessible cardinals " but not from ZFC + " There exist (at least) n inaccessible cardinals ". We prove similar results for infinitary rational relations accepted by 2-tape Büchi automata.