The Quest for a Formalization

Abstract

The difficulties that arose at the beginning of the twentieth century shook the foundations of mathematics and led to several fundamental questions: “What is an algorithm? What is computation? What does it mean when we say that a function or problem is computable?” Because of Hilbert’s Program, intuitive answers to these questions no longer sufficed. As a result, a search for appropriate definitions of these fundamental concepts followed. In the 1930s it was discovered—miraculously, as Gödel put it—that all these notions can be formalized, i.e, mathematically defined; indeed, they were formalized in several completely different yet equivalent ways. After this, they finally became amenable to mathematical analysis and could be rigorously treated and used. This opened the door to the seminal results of the 1930s that marked the beginning of Computability Theory.

A model of a system or process is a theoretical description that can help you understand how the system or process works.