Naive Axiomatic Class Theory: A Solution for the Antinomies of Naive Mengenlehre

TL;DR

This paper introduces a series of recursive axiom systems, including Naive Axiomatic Class Theory (NACT), to address classical antinomies in naive set theory, and explores their properties and applications in computational set theory.

Contribution

It constructs a hierarchy of partial recursive axiom systems derived from NACT to resolve classical paradoxes and discusses their medium classes and computational implications.

Findings

NACT systems form recursive axiom systems addressing antinomies.

Different variants of NACT exhibit distinct logical properties.

NSA-systems based on Not-SelfApplicability aid computational set theory.

Abstract

Since the axioms in (Consi-CoS) are not recursively enumerable, NACT* is no axiom system in the classical sense . Therefore we construct a series of partial systems which form a recursive axiom system too. Starting with the "dichotomic" systems NACT# and its variant NACT#4, we are going on to the "disjunctive" systems NACT+ and NACT+4, and eventually to NACT+Strat. After that we discuss the medium classes of these systems. Finally we present the inconsistent NSA-systems based on Not-SelfApplicability and explain their help for computational set theory.