Inhalt des Dokuments
Preprint 08-2017
Self-Inconsistency of set theory
Author(s) :
Michael Pfender
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 08-2017
MSC 2000
- 03B30 Foundations of classical theories
-
18-02 Research exposition
Abstract :
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. As a free-variable p. r. predicate, the consistency formula of T is decidable by T. Therefore, by Gödel's second incompleteness theorem, set
theories T turn out to be self-inconsistent, to derive their own
inconsistency formulae.
Keywords :
free variables, Skolem logic, iteration, primitive recursion, gödelisation, map code evaluation, objectivity, arithmetised equality, soundness, predicates, decidability, Gödel theorems, inconsistency provability