Foundations of Mathematics

In this chapter, we will delve deeper into the history of mathematical logic and introduce various concepts crucial for comprehending Gödel’s work. Our journey starts in Section 2.1 with a short visit to the late nineteenth century. Here, we will acquaint ourselves with Gottlob Frege, who not only made significant contributions to the development of modern logic but also stands as one of the tragic figures in the annals of science. Section 2.2 will discuss the contributions of Giuseppe Peano, in particular, his axiomatic foundation of natural numbers. The endeavors of Frege and Peano were pivotal in the life of our next protagonist, Bertrand Russell. Section 2.3 will derive Russell’s antinomy and explain why it damaged mathematics at its core. Subsequently, we will open up the monumental work that Gödel mentions in the title of his paper: the Principia Mathematica. Finally, Section 2.4 will discuss modern set theory and provide an overview of the various axiomatic systems invented to put mathematics on solid ground.