Introduction

With his 1929 thesis, Gödel delivers himself to us fully formed. He gives in it a definitive, mathematical treatment of the completeness theorem, but he also declares himself philosophically, unfolding the meaning of that theorem from a wider, rather mature, fully philosophical point of view.

Among the rewards of studying particularly its introductory remarks, are the following. First, light is shed on the timing of the first incompleteness theorem, construed as a response to Carnap as well as its possible genesis as a response to Brouwer; those remarks also add to our understanding of the separation of the completeness and categoricity concepts, which was emerging just then. In a few crucial places, the remarks can strike the modern reader as peculiar. The view taken here is that these peculiarities are interesting and important, and therefore they are treated at length in this chapter. The introductory remarks were never included in the publication based on the thesis, and indeed, Gödel would not publish such unbuttoned philosophical material until 1944, with his On Russell's Mathematical Logic.

The Introduction to Gödel's Thesis: Different Notions of Consistency

The set of remarks we first consider occur in the first paragraph of Gödel's 1929 thesis, and address the issue of whether consistency is a ground for existence: