# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1908.02539