Consistency Decision

TL;DR

This paper demonstrates that classical set theory, under the assumption of ω-consistency, can decide its own consistency formula using primitive recursive maps, leading to the conclusion of its ω-inconsistency.

Contribution

It introduces a method to decide set theory's consistency formula via primitive recursive maps, challenging traditional views on set theory's consistency and incompleteness.

Findings

Set theory's consistency predicate is decidable within set theory.

Under ω-consistency, set theory's own consistency formula can be decided.

Classical set theory is shown to be ω-inconsistent under these assumptions.

Abstract

The consistency formula for set theory can be stated in terms of the free-variables theory of primitive recursive maps. Free-variable p. r. predicates are decidable by set theory, main result here, built on recursive evaluation of p. r. map codes and soundness of that evaluation in set theoretical frame: internal p. r. map code equality is evaluated into set theoretical equality. So the free-variable consistency predicate of set theory is decided by set theory, {\omega}-consistency assumed. By G\"odel's second incompleteness theorem on undecidability of set theory's consistency formula by set theory under assumption of this {\omega}- consistency, classical set theory turns out to be {\omega}-inconsistent.