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Generalized Von Neumann Universe and Non-Well-Founded Sets

Eugene Zhang

TL;DR

The paper proposes the total universe, an enlarged von Neumann hierarchy that explicitly includes non-well-founded sets (infinitons, semi-infinitons, quasi-infinitons) via limits of finite structures and formulas. It develops a rigorous framework using limit theories (both in infinitary logics and set-theoretic language) to define infinitely generated sets and their types, then constructs the total universe T with new inf-generation operators and a spectrum of power-set operations. EZF, an extended ZF-like theory, is introduced to accommodate both WF and NWf sets, and the Russell paradox is shown to be avoidable within this framework by distinguishing infiniton classes and their hierarchies. Overall, the work argues that Regularity is not necessary for consistency with ZF and that the total universe provides a robust, paradox-free foundation integrating non-well-founded sets with well-founded ones. The approach offers a new, rigorous route to study non-well-founded phenomena using limits, spectrum-based construction, and an extended axiomatic base.

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 is invalid in defining well-founded sets and wrong in any system consistent with ZF set theory.

Generalized Von Neumann Universe and Non-Well-Founded Sets

TL;DR

The paper proposes the total universe, an enlarged von Neumann hierarchy that explicitly includes non-well-founded sets (infinitons, semi-infinitons, quasi-infinitons) via limits of finite structures and formulas. It develops a rigorous framework using limit theories (both in infinitary logics and set-theoretic language) to define infinitely generated sets and their types, then constructs the total universe T with new inf-generation operators and a spectrum of power-set operations. EZF, an extended ZF-like theory, is introduced to accommodate both WF and NWf sets, and the Russell paradox is shown to be avoidable within this framework by distinguishing infiniton classes and their hierarchies. Overall, the work argues that Regularity is not necessary for consistency with ZF and that the total universe provides a robust, paradox-free foundation integrating non-well-founded sets with well-founded ones. The approach offers a new, rigorous route to study non-well-founded phenomena using limits, spectrum-based construction, and an extended axiomatic base.

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 is invalid in defining well-founded sets and wrong in any system consistent with ZF set theory.
Paper Structure (22 sections, 82 theorems, 121 equations, 4 figures)

This paper contains 22 sections, 82 theorems, 121 equations, 4 figures.

Table of Contents

  1. Preliminaries
  2. Introduction
  3. Problems in Von Neumann Universe
  4. Correct Rank in Von Neumann Universe
  5. Unpacking Operator and Nullity
  6. Membership Dimension
  7. Limit of Structures and Formulas
  8. Limit in General Language
  9. Limit in Language of Set Theory
  10. Non-Well-Founded Sets
  11. Introduction
  12. Infinitely Generated Set
  13. Infiniton
  14. Semi-Infiniton
  15. Quasi-Infiniton
  16. ...and 7 more sections

Key Result

Lemma 1.2

$V$is well-founded and no set in $V$ is non-well-founded.In this paper, an existing theorem in set and model theory is listed as a proposition, while a (mainly) new result is proved as a theorem. Lemmas, corollaries, conclusions and axioms can have both new and existing results.

Figures (4)

  • Figure 1: Diagrams of infinitons and a set of infinitons.
  • Figure 2: Diagram of a semi-infiniton.
  • Figure 3: Diagram of a quasi-infiniton.
  • Figure 4: Diagram of the total universe.

Theorems & Definitions (135)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Example 1.10
  • ...and 125 more