The Incompleteness Theorems, their content and their meaning
Dick de Jongh
Abstract:
In the year 2006, 100 years after G~del'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. In Section 2, a rough version of the theorems and the in my
opinion main achievements of G~del 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Sections 3 and 4, the historical content of the theorems is sketched,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
G~del, 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.