Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

We prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like [Formula: see text] [Formula: see text] “There exist (at least) [Formula: see text] inaccessible cardinals”, for integers [Formula: see text]. In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set [Formula: see text] of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.