ZFK := ZFC with a Complement, or: Hegel and the Synto-Set-Theory

TL;DR

This paper explores minimal modifications to ZF set theory to include a complement axiom, proposing ZF'' as the minimal extension and ZFK as the simplest theory with a universal set, connecting to Hegelian philosophy.

Contribution

It identifies ZF'' as the minimal modification of ZF with a complement axiom and introduces ZFK as the simplest set theory with a universal set.

Findings

ZF'' is the minimal modification of ZF with a complement axiom.

ZFK is the simplest set theory with a universal set.

The systems relate to philosophical ideas from Hegel.

Abstract

What is the slightest modification of ZF to add a complement-axiom? The answer in my Ph.D. thesis 1971 was ZF'': Zermelo-Fraenkel with replacement for only well-founded domains and an omega-axiom. In 1974, Alonzo Church published a similar system, as did Urs Oswald in 1976. In his 1976 Ph.D. thesis, E. Mitchell also designed, a system very similar to ZF''. In this article we argue that ZF'' is the slightest modification of ZF among all these systems and that its successor ZFK is the simplest set theory with a universal set at all.