## A Method in Proofs of Undefinability: With Applications to Functions in the Arithmetic of Natural Numbers |

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### Contents

A method in proofs of undefinability | 14 |

Addition and multiplication in the formalized | 25 |

Results concerning equational undefinability | 41 |

1 other sections not shown

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### Common terms and phrases

arithmetic of natural arithmetical meaning atomic formula axiom system binary operation constants binary relation ck with respect closure property compatible consistent extension consistent theory containing constant individual constant constant is undefinable constants with respect containing all sentences containing the non-logical Corollary definable definition problem equational definition equational undefinability equationally and validly essentially undefinable equationally essentially undefinable explicitly explicit definition explicit undefinability explicitly and compatibly explicitly and validly extended fc-ary operation constant finitely axiomatizable follows formalized arithmetic interpretation introduced a notion Lemma logical constants method in proofs minimal extension model Mo natural numbers o(xi obvious one-model method operation constant individual Padoa's method pm-i pm(a pm(x PROOFS OF UNDEFINABILITY property for pi recursive set relation constant respect to T2 respect to theory sentence being valid sequence pi shape subtheory symbols Ta(S terms of models Tm(a undecidable valid sentences validly or compatibly xk+i