Ontologia matemática: de ZFC ao estruturalismo de Benacerraf

Abstract

This work investigates the ontology of mathematical objects, analyzing the transition from the foundation in Set Theory to the contemporary structuralist perspective. The first part is dedicated to the systematization of the Zermelo-Fraenkel axiomatic system (ZFC), demonstrating how its structure is essential to avoid logical paradoxes, such as Russell's, and to allow the internalization of natural numbers as sets through the constructions of von Neumann and Zermelo. Subsequently, the research analyzes the challenge proposed by Paul Benacerraf, who uses the technical divergence between these models to expose the arbitrariness in identifying numbers with specific sets, which gives rise to the so-called "junk theorems". The study concludes, based on Benacerraf's thesis, that numbers do not possess a "substance" or internal constitution, and must be understood not as independent objects, but as relational positions within an abstract structure.