I should have explained that Kurt Gödel and the Foundations of Mathematics is divided into three main parts, ‘Historical Context’, ‘A Wider Vision: the Interdisciplinary, Philosophical and Theological Implications of Gödel’s Work’, and ‘New Frontiers: Beyond Gödel’s Work in Mathematics and Symbolic Logic’. There are no less than seven papers in the first part left to talk about, and I’m not really the best person to comment on their historical content. But on we go.
Next, then, is Karl Sigmund on “Gödel’s troubled relationship with the University of Vienna based on material from the archives as well as on private letters”. The bald outlines will be familiar e.g. from Dawson’s biography, but there’s a lot of new detail. This is a nicely readable paper about the facts of the case, but I don’t think that there’s anything here that sheds new light on Gödel’s intellectual progress.
The fifth paper is ‘Gödel’s Thesis: an Appreciation’ by Juliette Kennedy. This concentrates on the philosophical Introduction of Gödel’s 1929 thesis on completeness (the material which wasn’t included in the 1930 published version), and in particular on his remarks about whether consistency implies existence. So part of Kennedy’s paper is in effect an elaboration of what Dreben and van Heijenoort say on p. 49 of their intro in the Collected Works when they point out that Gödel’s remarks are a bit misleading. She goes on also to urge that that there is evidence that Gödel was already, in 1929, thinking about incompleteness for theories (also not really a new suggestion). Unless I’m missing something — and Kennedy isn’t ideally clear — this also doesn’t really add much to our understanding. [That’s too brisk — see the Comments.]
The following paper is a typically lucid piece by Solomon Feferman, on the Bernays/Gödel correspondence, and as always is well worth reading if you are interested in the history here. Feferman has already introduced the correspondence in the Collected Works: but this piece concentrates at greater length on the theme of ‘Gödel on Finitism, Constructivity, and Hilbert’s Program’. As he writes,
There are two main questions, both difficult: first, were Gödel’s views on the nature of finitism stable over time, or did they evolve or vacillate in some way? Second, how do Gödel’s concerns with the finitist and constructive consistency programs cohere with the Platonistic philosophy of mathematics that he supposedly held from his student days?
Re the first question, Feferman wrote in CW of Gödel’s ‘unsettled’ views on the upper bound of finitary reasoning. Tait has dissented. But in fact,
Tait says that the ascription of unsettled views to Gödel in the correspondence and later articles “is accurate only of his view of Hilbert’s finitism, and the instability centers around his view of whether or not there is or could be a precise analysis of what is ‘intuitive”’ (Tait, 2005, 94). So, if taken with that qualification, my ascription of unsettled views to Gödel is not mistaken. As to Gödel’s own conception of finitism, I think the evidence offered by Tait for its stability is quite slim …
On the second question, of why Gödel should have devoted so much time to thinking about a project with which he was deeply out of sympathy, Feferman writes
Let me venture a psychological explanation … : Gödel simply found it galling all through his life that he never received the recognition from Hilbert that he deserved. How could he get satisfaction? Well, just as (in the words of Bernays) “it became Hilbert’s goal to do battle with Kronecker with his own weapon of finiteness,” so it became Gödel’s goal to do battle with Hilbert with his own weapon of the consistency program. When engaged in that, he would have to do so – as he did – with all seriousness. This explanation resonates with the view of the significance of Hilbert for Gödel advanced in chapter 3 of Takeuti (2003) who concludes that … [Gödel’s] “academic career was molded by the goal of exceeding Hilbert.”
Sounds plausible to me!
Thanks for looking at my paper! It is an honor to be reviewed in “Logic Matters.”
If I may explain what I attempted to do in the paper, this was two things: first, to connect the introductory remarks in Godel’s thesis to his phrasing of the incompleteness theorem in his 1930 Konigsberg announcement of them. This would weaken the tie of these theorems to Hilbert, and strengthen their tie to Carnap (and on other grounds, to Brouwer). In particular, the Incompleteness Theorems refute the idea of there being a completeness theorem for second order logic, as Carnap thought, something which I think is not sufficiently emphasized in the literature.
Secondly I wanted to make vivid the general surrounding context, in particular contemporary notions of completeness, categoricity and the distinction between first and second order logic. I hoped that in the last few paragraphs of the paper I explain why the first order concept of logic is so powerful.