Generalized Von Neumann Universe and Non-Well-Founded Sets

2023/04/02 by Eugene Zhang, Zhang, Eugene
Computer Science · Mathematics · Psychology · #03C55 #03C75 #03E99 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Philosophy and Theoretical Science

paper · pdf · doi:10.48550/arxiv.2304.00581

openalex publication_date 2023/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, a generalized version of the von Neumann universe known as the total universe is proposed to formally introduce non-well-founded sets that include infinitons, semi-infinitons and quasi-infinitons in Russell's paradox. All three infinitons are part of infinitely generated sets that are generators of non-well-founded sets. Combining the well-founded sets with the non-well-founded sets, the total universe is a model of ZF minus the axiom of regularity and free of Russell's paradox. The axiom of regularity can not define the well-founded sets and is invalid in any system consistent with ZF set theory.

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