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Self-Inconsistency of set theory

Pfender, Michael

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

The consistency formula for set theory T e. g. Zermelo-Fraenkel set theory ZF, can be stated in form of a free-variable predicate in terms of the categorical theory PR of primitive recursive functions/maps/predicates. Free-variable p. r. predicates are decidable by T, key result. Decidability is built on recursive evaluation of p. r. map codes and soundness of that evaluation into theory T : internal, arithmetised p. r. map code equality is evaluated into map equality of T. In particular, thefree-variable p. r. consistency predicate of T is decidable by T. Therefore, by Gödel’s second incompleteness theorem, set theories T turn out to be self-inconsistent.