O "verdadeiro" fundamento da matemática elementar

Abstract

This work aims to study the beginning and development of axiomatic systems, focusing mainly on set theory, created by George Cantor. The goal is to define axiomatic systems and show some paradoxes presented by the so-called “naive” set theory, where some examples of paradoxes and definitions in other theories will also be given to explain the scope and relevance of the content treated here, and how the axiomatization created by Zermelo and Fraenkel is able to avoid these paradoxes. Also, the equivalence between the Axiom of Choice, Zermelo’s theorem and Zorn’s Lemma will be presented. We offer a special section to demonstrate the surprising equivalence between the Axiom of Choice and the statement that every vector space has a basis. Finally, we give some additional sections talk about Gödel’s incompleteness theorem and show statements that cannot be proven in the ZFC system.