Comments for Logic Matters

Comment on Carnap and the Diagonalization Lemma (Continued) by Peter Milne

Peter Milne — Mon, 06 Feb 2012 19:23:02 +0000

I’ve now had the chance to check this. The Diagonal Lemma/Fixed-Point Theorem isn’t stated as such in Tarski, Mostowski and Robinson but is easily derived from what the three say in section 2, “Definability in arbitrary theories”, in the jointly authored second paper (see pp. 44-47 of the Dover reprint). But since interested there only in decidability, the steps in the derivation aren’t even hinted at. So who did first make that easy derivation? One of Tarski’s students?

[In his 'Theories incomparable with respect to relative interpretability' of 1962 Richard Montague refers to the diagonal lemma for Robinson Arithmetic as "well known" and refers to "the familiar proof".]

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by Peter Milne

Peter Milne — Mon, 06 Feb 2012 19:03:47 +0000

The Ishi Press reprint is some sort of photo-reprint of a pre-1971 edition, an early edition I think. It contains a new forward by Michael Beeson.

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by scott

scott — Thu, 02 Feb 2012 11:47:28 +0000

Thank you very much! I just ordered a copy from abebooks.

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by Peter Smith

Peter Smith — Thu, 02 Feb 2012 10:58:32 +0000

In footnote 1 of his short piece ‘The writing of Introduction to Mathematics‘ (in Thomas Drucker, ed., Perspectives on the History of Mathematical Logic (Birkhäuser 1991), Kleene says

There has been no revision [between the nine reprintings up to 1988]–only a few equal-space changes, two notes added in open spaces at ends of chapters, and the updating of eleven references which in 1952 were “to appear,” as is indicated on p. vi of the sixth (1971) and later reprints.

So that suggests you can’t go far wrong with any printing, and any from 1971 should be the same.

By the way, there are a lot of inexpensive second-hand copies on abebooks.com

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by scott

scott — Thu, 02 Feb 2012 08:49:22 +0000

Professor Smith, I wonder if you wouldn’t mind sharing any knowledge you might have on the various editions and reprints of Kleene’s Introduction to Metamathematics. I’m trying to get my hands on a copy and while this isn’t itself terribly difficult, what I do find a bit confusing is that Kleene’s book seems to have passed through several publishing houses. Most recently, the book was published by Ishi Press (and according to most accounts, this is essentially an on-demand photocopy). There is a printing from about 1980 on North Holland Press and current copies of this also seem to be “print on demand.” And it is still possible to find used copies of the third and fourth printings of what at least appears to be the original text published by D. Van Nostrand.

My question: As far as you know, is there a preferred text? (By which I mean a preferring printing on a particular press?) I’d be extraordinarily grateful to you for any insight you might have on this.

By the way, I’m very much looking forward to the second edition of your Introduction to Godel’s Theorems. It’s a wonderful book and I’ll be using it to teach an advanced undergraduate course in logic that I am developing for next year.

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by David Auerbach

David Auerbach — Wed, 25 Jan 2012 17:30:49 +0000

I was having similar thoughts but with some historical threads (and this may be more like real history of the subject than you want to get into). The Smullyan material comes out of Post Canonical Systems and poor Post is often left out of these accounts. And PCSs are sort an ideal bridge between explicitly recursion theoretic accounts and the who-needs-numbers-we’re-talking-syntax accounts. (I guess the idea here is that from a PCS point of view you don’t analyze formalisms in terms of recursiveness–it’s really the other way around.) The other Smullyan thread, the one that’s more apparent in his later books (but also in some very cute technical articles) is the drive to a certain level of abstraction. I want to distinguish abstraction from generalization. Brute generalization is just, for instance, where you prove G1 for all extensions of Q. Abstraction is what Smullyan does in his GIT book or what happens when we study the modal logic provability. But those moves were all already happening in the 1930s; the HB derivability conditions being the obvious example. (I count those as initiating the abstraction route.) Feferman’s great paper (The Arithmetization of Metamathematics…) sort of converts abstraction to generalization, by (sort of) finding a way to specify the range of formalisms that do satisfy the HB derivability conditions. And that’s my potted history ramble.

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by Peter Smith

Peter Smith — Wed, 25 Jan 2012 09:39:02 +0000

Yes — I have nothing but admiration for Smullyan! I’ll certainly be discussing his Theory of Formal Systems later in the notes, as well as some of his other wonderful books.

Comment on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943 by Marco

Marco — Wed, 25 Jan 2012 08:50:08 +0000

I wish to suggest an alternative proof related to Godel’s results, which goes back to Raymond Smullyan in his “Theory of Formal Systems”,
This is an excerpt from the introduction:
“Godel’s program of arithmetizing syntax
is accomplished in a new manner; no appeal is made to primitive recursive
function theory, prime factorization, theory of congruences or the Chinese
remainder theorem. A by-product of this approach (which was undertaken
primarily out of considerations of elegance) is that improved normal form
theorems are obtained.”

Comment on Carnap and the Diagonalization Lemma (Continued) by Stefan Roski

Stefan Roski — Thu, 19 Jan 2012 00:09:45 +0000

A small remark: If you know the English translation of Logische Syntax, you essentially also know Carnap1934a. This paper contains material that Carnap could not include in the original German version, as the publisher wanted to keep the book short. It is, however, contained in the English translation (cf. the preface to the latter).

Comment on Carnap and the Diagonalization Lemma (Continued) by Peter Smith

Peter Smith — Tue, 17 Jan 2012 08:11:23 +0000

Thanks for all this, Peter.

I don’t know Carnap 1934a either — so I can’t comment on that.

But as it chances, I was reading Rosser 1939 just yesterday. His Lemma 1 seems clearly to be a statement of the semantic Diagonalization Equivalence (as I called it) rather than the syntactic Diagonalization Lemma (and that’s all his proof delivers — it’s all about what can be expressed, not what can be numeralwise represented (or whatever your favourite jargon is).

Gaifman’s ‘ function that maps every sentence χ to α (‘χ’) equivalent to it in T’ is ambiguous between whether it is semantic equivalence or provable equivalence which is in question. His remark about what is trivial to extract doesn’t seem that apt to the semantic Diagonalization Equivalence (once you’ve cracked Gödel’s rather obtuse mode of presentation in 1931). But true, it is perhaps slightly less trivial to extract the Diagonalization Lemma proper, but then I can’t see that Carnap got it clearly either.

And yes, I agree entirely about Gödel’s added footnote.