A Note on the Possibility of Self-Reference in Mathematics

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.