Paradigms-Shift in Set Theory

TL;DR

This paper discusses a paradigm shift in set theory from ZFC to NACT, proposing systems where all non-proper classes are sets, using non-monotonic logic to create large set universes based on philosophical principles.

Contribution

It introduces new set-theoretic systems like NACT variants that aim to incorporate all non-proper classes as sets, driven by philosophical motivations rather than purely mathematical ones.

Findings

Development of NACT-based systems producing large set universes

Introduction of NACT+NFUM-closed and related approximations

Proposals for systems unifying various set theories through philosophical principles

Abstract

In this article the author claims that there is a paradigm shift from ZFC to NFUM and further to NACT - due to philosophical reasons, not mathematical ones. The goal is to construct systems where every "Not-Properclass" is a set! With help of Non-Monotonic Logic, the consistent systems NACT-MoonW, NACT*W, and NACT-SunW are producing "largest possible universes" of sets. Using self-evident philosophical principles, three approximations are suggested: NACT+NFUM-closed (to NACT*), NACT&ZFC4+(GCH) (also to NACT*) and NACT-NFUM (to NACT-Sun). Also the system NACT[&ZFC4-closed]+(FCA) is considered. NFUM = NFU with (AC) and measurable properclass Ord. NFUM-closed is NFUM where the set-constituting formulas Ai in the set operator need not only to be stratified but also to be made parameter-free (i.e., have only x as free variable over which the set is comprehended). In other formulas, free variables are allowed. Keywords: Set Theory, ZFC, Naive Set Theory, Predicate-Extension, Church Schema of Comprehension, NF, NFUM, Stratification, Universal Sets, Eliminative Class Theory, NBG, NACT (= Naive Axiomatic Class Theory), NACT-Sun, NACT-Moon, NACT-Star, NACT+NFUM-closed, ZFCK, NACT+NFUM, (GCH), (FCA) [= Finite or Countable Anti-Thesis], NACT+ZFC4+(GCH), NACT[+ZFC4-[closed]+(FCA).