The Incompleteness Theorems, their content and their meaning

In the year 2006, 100 years after Gödel’s birth, it is time to think about the lasting values of his achievements. In this paper I will be concerned with his most famous results, his two incompleteness theorems1 [9]. In Section 2, a rough version of the theorems and the in my opinion main achievements of Gödel by these theorems are given. In Sections 3 and 4, the historical content of the theorems is sketched, in Section 5, I define what a formal system is. In Sections 6 and 7, I describe the content of the first incompleteness theorem somewhat more precisely and give the main ideas contained in the proof. The second incompleteness theorem is treated in section 8. Two developments after Gödel, Provability logic and Feferman’s arithmetization of metamathematics are considered in Section 9. I conclude with some remarks on Lucas’ and others’ attempts to derive philosophical conclusions concerning human beings vs machines from the first incompleteness theorem.