Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity

TL;DR

This paper argues that Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is inconsistent due to contradictions in infinite set cardinalities, impacting computational complexity theory.

Contribution

It demonstrates the inconsistency of ZFC and explores its implications for computational complexity, a novel connection not previously established.

Findings

ZFC contains contradictions in infinite set cardinalities

Peano arithmetic is inconsistent under ZFC assumptions

Implications for computational complexity theory are discussed

Abstract

This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.