A Note on the Possibility of Self-Reference in Mathematics

TL;DR

This paper explores how self-referential propositions in a meta-model of ZF can lead to inconsistencies, linking these findings to interpretations of Gödel's incompleteness theorem and discussing broader implications.

Contribution

It introduces an interpretation of self-reference in a meta-model of ZF and analyzes its implications for consistency and foundational issues in mathematics.

Findings

Self-referential propositions can cause inconsistency in the meta-model N*.

Some legitimate mathematical propositions become problematic under this interpretation.

The work connects these issues to interpretations of Gödel's first incompleteness theorem.

Abstract

In this paper we propose an interpretation for self-referential propositions in a "meta-model" N* of ZF. This meta-model N* is considered as an informal model of arithmetic that mathematicians often use when working with number theory. Specifically, we assume that within this meta-model, the axiom system ZF is applied, interpretations for sentences can be offered, and natural language can be used. We show that under the proposed interpretation, some types of self-referential propositions that are considered legitimate in mathematics turn N* into an inconsistent model, and examine the connection of this result to a certain interpretation of godel's first incompleteness theorem. Some general problems which follow from the above discussion are then addressed.