Here’s a short new paper by Kripke on proving the first incompleteness theorem.
First impression is that this gives an interesting little twist rather than significant novelty: but fun all the same!
Here’s a short new paper by Kripke on proving the first incompleteness theorem.
First impression is that this gives an interesting little twist rather than significant novelty: but fun all the same!
Pedagogically it’s a really useful improvement though. The diagonal lemma in its current form causes many tears.
Somewhere and long ago I wrote (and attributed to Kripke but as folklore):
Let PA⋆ be this language together with new constants c1,c2, c3, . . . . Let φ1(x), φ2(x), . . . be all the wffs of PA⋆ with x as sole free variable. Interpret ci as follows:
I(ci) = φi(ci)
Lemma 5.1 (Fixed Point Lemma) For every φi there is a wff S and a constant c
such that;
I(φi(c) ≡ S) = TRUE and I(c) = S.
ETC.
Kripke’s reference to Smullyan, but not to a paper, made me think of Smullyan’s little known (?) “Chameleonic Languages”. (Synthese 1984).
How is GWT coming along? Can’t wait to see the printed version!
Kripke’s writings on the incompleteness theorems are some of my favorites of his; I especially love his arguments in his The Road to Gödel, it raised a lot of interesting thoughts for me regarding the semantic paradoxes and what does it really mean to say that two of them are the same, and there is interesting work connecting that question to graph theory, especially in regards to Yablo’s paradox and other ‘backwards omega sequence’ paradoxes, as Yablo and Rayo call them.
Kripke was teaching a class based on his Elementary Recursion Theory and its Application to Formal Systems manuscript last semester and Romina Padro told me that the plan is to refine it and publish it, here’s hoping it happens sooner rather than later. Personally, I’ve spent the past four years waking up every day hoping to hear news that his second collected works volume has a release date. I’d be interested in your review of EFT and where it fits into the Logic guide; I like the nonstandard way he presents the subject, as a second or third course, as well as the many asides on the philosophical problems related to computability and incompleteness he gives.