[Paper Review] Brief Lecture Notes on Self-Referential Mathematics, and Beyond

Elemér E Rosinger|ArXiv.org|May 2, 2009

Computability, Logic, AI Algorithms5 references3 citations

TL;DR

This paper introduces a consistent extension of Zermelo-Fraenkel set theory by replacing the Axiom of Foundation with an Anti-Foundation Axiom (AFA), enabling the existence of self-referential sets while preserving consistency if ZFC is consistent. It argues that self-reference is not inherently paradoxical and highlights the practical use of inconsistent systems in digital computing, advocating for a broader mathematical framework embracing self-reference and contradiction.

ABSTRACT

Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent Mathematics.

Motivation & Objective

• To propose a consistent extension of standard set theory by replacing the Foundation Axiom with an Anti-Foundation Axiom (AFA), allowing self-referential sets.
• To challenge the negative connotation of 'vicious circle' in set theory by reinterpreting self-reference as a fundamental and non-paradoxical concept in mathematics and human thought.
• To argue that inconsistent mathematics—exemplified by digital computers operating under contradictory axioms—is already in widespread practical use.
• To advocate for a broader mathematical framework that unifies self-referentiality and inconsistency, suggesting vast new realms of mathematical exploration.
• To clarify foundational distinctions between sets, ur-elements, and classes, and to formalize the role of self-referential definitions within a consistent axiomatic system.

Proposed method

• Replacing the Foundation Axiom (FA) in ZFC with the Anti-Foundation Axiom (AFA), which asserts that every flat system of equations has a unique solution.
• Defining a hierarchy of entities: sets (SET), ur-elements (U), and classes (CLASS), with ur-elements being non-sets that contain no elements.
• Using the AFA to allow solutions to equations like x = {x}, thereby generating self-referential sets that are not well-founded under FA.
• Formalizing the distinction between sets, proper classes, and ur-elements, ensuring that SET and U are proper classes not belonging to SET.
• Introducing the Strong Axiom of Plenitude to generate new ur-elements from existing sets and subsets of ur-elements, ensuring uniqueness and injectivity.
• Demonstrating that ZFA (ZFC minus FA plus AFA) is consistent if ZFC is consistent, using model-theoretic reasoning.

Experimental results

Research questions

• RQ1 Can self-referential sets be consistently incorporated into standard set theory without violating foundational principles?
• RQ2 What is the mathematical and philosophical significance of replacing the Foundation Axiom with an Anti-Foundation Axiom?
• RQ3 How does the use of self-referential definitions in set theory relate to historical and cognitive traditions of self-reference in human thought?
• RQ4 To what extent are real-world systems like digital computers already based on inconsistent mathematics, and what does this imply for foundational mathematics?
• RQ5 What new mathematical structures and realms might become accessible by unifying self-referentiality and inconsistency in formal systems?

Key findings

• The Anti-Foundation Axiom (AFA) ensures that every flat system of equations has a unique solution, thereby enabling the existence of self-referential sets such as x = {x}.
• The resulting set theory ZFA is consistent if ZFC is consistent, meaning that self-referential sets can be added without introducing logical contradictions.
• Self-referential sets are not inherently paradoxical; the term 'vicious circle' is misleading and stems from a narrow historical view centered on Russell’s paradox.
• Digital computers operate under a contradictory system: they satisfy the Peano Axioms but also the Machine Infinity Axiom (M + 1 = M for very large M), which is inconsistent with Peano arithmetic.
• The existence of such practical inconsistency in computing suggests that inconsistent mathematics is not only viable but already essential in modern technology.
• The paper proposes that a new mathematical frontier lies in combining self-referentiality and inconsistency, potentially unlocking vast new domains of mathematical exploration.