When I wrote my Gödel book I did a lot of it from memory (on the principle that if I had to reconstruct proofs without too much cheating, the discipline of doing so would help me to explain the proofs). But lately, as I slowly work away at the second edition, I’ve been (re)reading some of the old literature around and about the incompleteness theorems.
I’ve had a copy of Smullyan’s Theory of Formal Systems (1961) for thirty years, but to be honest I don’t think I’ve ever read it properly before. And it does date from that time when many maths books were photo-printed from typewritten originals, and looked exceedingly uninviting. Moreoever, in some ways, this volume has been superseded by Smullyan’s Gödel’s Incompleteness Theorems (1992).
But still, as I’ve found with great enjoyment, the earlier book remains quite terrific stuff. It is just so very elegant and insightful in developing Post’s ideas on formal systems and in excavating one basic line of ideas underlying incompleteness proofs (I plan to write up some notes in due course). Impressive indeed — even more so when you remember the book is essentially Smullyan’s doctoral thesis. Still very worth reading after all this time.
(And how many philosophy books, as opposed to logic books, still stand up so well after fifty years?)
Professor Smith,
Just wondering: Has there been any headway on notes for Smullyan’s Formal Systems? I’m a graduate student in logic and I bought a copy of the book and could benefit from a tutorial in Smullyan’s treatment of formal systems. — Andrew
I’m afraid I got distracted from that project. One day …
I will continue to hope, then! :)
Something worth noting is that Smullyan has recently written another book dealing with Godel, fixed points, etc, “The Godelian Puzzle Book.” It’s clarity is rather astounding, and it manages to go from Knights & Knaves type Puzzles to Godel’s first incompleteness theorem for Peano arithmetic. He covers a lot of territory in a very accessible manner. Definitely worth checking out!
G. Kreisel’s review of the book for Mathematical Reviews is worth reading.
Thanks for the reminder: Kreisel’s review is here.
The last book of Smullyan’s saga on Godel’s results and beyond,
namely Diagonalization and Self-Reference, is also a good reference
to understand the nature of formal systems in Post’s style.
There are some relevant improvements with respect to the earlier work.
A wonderful book. I haven’t read it since 1976 or so. Prompted by your note, I just pulled it down off the shelf. The front cover is missing (but the back, loose, is there in all its glorious orange), the rest only slightly yellowed and securely sewn in signature. There are marginal notes in it by, it seems, the younger and smarter me. Jeroslow’s work was a neat follow-on. I do think that Post Canonical systems are the appropriate explication of a formal system.