Preprint 04-2020

Inconsistency of set theory via evaluation

Author(s) : Michael Pfender , C.C. Nguyen, J. Sablatnig

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 04-2020

MSC 2000

03B10 Classical first-order logic

03F40 Gödel numberings in proof theory

18A15 Foundations, relations to logic and deductive systems

Abstract :
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 abstrac- tion 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 the- ory PR via Complexity Controlled Iteration with witnessed termination of the iteration, witnessed termination by avail- ability of Hilbert’s iota operator in set theory. Objectivity of that evaluation yields inconsistency of set theory T by a liar (anti)diagonal argument.

Keywords : Classical first-order logic, Goedel numberings and issues of incompleteness, Foundations, relations to logic and deductive systems