Resultados sobre la axiomatización de las teorías de conjuntos ZF y NBG

Abstract

[en] The goal of this project is to study the axiomatic systems typically used to present two versions of axiomatic set theory: ZF and NBG. Specifically, we will analyze whether some of their axioms are independent of the other ones or consistent with the remaining, being always assumed that the initial theory is consistent. Furthermore, we will also prove various theorems in order to answer the question about whether it is possible to axiomatize ZF or NBG with a finite amount of axioms, being the answer negative for the first and affirmative for the second. We will conclude with a result that asserts that NBG is a conservative extension of ZF, that is, both prove the same theorems regarding sets only. To that end, we will have to first introduce some basic set theory preliminaries, study the Axiom of Foundation and the Cummulative Hierarchy and define the concepts of relativization and absoluteness for formulas.