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Inconsistency of set theory via evaluation

Pfender, Michael; Nguyen, C.C.; Sablatnig, J.

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

We introduce in an axiomatic way the categorical theory PR of primitive recursion as the initial cartesian category with Natural Numbers Object. This theory has an extension into constructive set theory S of primitive recursion with abstraction of predicates into subsets and two-valued (boolean) truth algebra. Within the framework of (typical) classical, quantified set theory T we construct an evaluation of arithmetised theory PR via Complexity Controlled Iteration with witnessed termination of the iteration, witnessed termination by availability of Hilbert s iota operator in set theory. Objectivity of that evaluation yields inconsistency of set theory T by a liar (anti)diagonal argument.