Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

  • Olivier FinkelEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)


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 \(T_n =:\mathbf{ZFC} +\) “There exist (at least) n inaccessible cardinals”, for integers \(n\ge 0\). 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 \(\mathbb {Z}^{3\times 3}\) of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.


Automata and formal languages Logic in computer science Finite words Context-free grammars 2-tape automaton Post correspondence problem Weighted automaton Finitely generated matrix subsemigroups of \(\mathbb {Z}^{3\times 3}\) Models of set theory Incompleteness theorems Large cardinals Inaccessible cardinals Independence from the axiomatic system “ZFC + there exist n inaccessible cardinals” Independence from Peano Arithmetic 


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu - Paris Rive GaucheCNRS et Université Paris 7ParisFrance

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