I think I might be going to regret this. I’ve just realized that there are over 25 “Big Books” on math logic on my shelves that could reasonably warrant a mention in the planned chapter in the Teach Yourself Logic Study Guide. So no promises to get everything finished quickly! Still, the chronologically next entry writes itself pretty easily. So let’s get it out of the way.
Joseph R. Shoenfield’s Mathematical Logic (Addison-Wesley, 1967: pp. 334) has, over the years, also been much recommended and much used: it is officially intended as ‘a text for a first-year [maths] graduate course’. This is hard going, a significant jump up in level from Mendelson, though often the added difficulty in mode of presentation seems to me not to be necessary. The book can probably only be recommended to hard-core mathmos who already know a fair amount and can cherry-pick their way through the book. It does have heaps of hard exercises, and some interesting technical results are in fact buried there. But whatever the virtues of the book, they certainly don’t include approachability or elegance or student-friendliness in the early chapters.
In a bit more detail, Chs. 1–4 cover first order logic, including the completeness theorem. It has to be said that the logical system chosen is rebarbative. The primitives are $latex \neg$, $latex \lor$, $latex \exists$, and $latex =$. Leaving aside the identity axioms, the axioms are the instances of excluded middle and instances of $latex \varphi(\tau) \to \exists\xi\varphi(\xi)$, and then there are five rules of inference. So this neither has the cleanness of a Hilbert system not the naturalness of a natural deduction system. Nothing is said to motivate this horrible choice as against others.
Ch. 5 is an introduction to some model theory getting as far as the Ryll-Nardjewski theorem. But this will done far too rapidly for most readers (unless you are using it as a terse revision course).
Chs. 6–8 cover the theory of recursive functions and formal arithmetic. Schoenfield defines the recursive functions as those got from an initial class by composition and regular minimization. As elsewhere, ideas are presented in a take-it-or-leave-it spirit, and no real motivation for the choice of definition is given (and e.g. the definition of the primitive recursive functions is relegated to the exercises). Unusually for a textbook at this sort of level, the discussion of recursion theory in Ch. 8 goes far enough to cover a Gödelian ‘Dialectica’-style proof of the consistency of arithmetic, though the presentation is not particularly accessible.
Ch. 9 on set theory is perhaps the book’s real original raison d’être. It is a quarter of the whole text and was (if I recall right) the first extended textbook presentation of Cohen’s independence results via forcing, from four years previously. The treatment also touches on large cardinals. This was surely an admirable achievement in its time: but it is equally surely not now the place to start with set theory in general or forcing in particular, given the availability of later presentations. [Or at least, that was my first thought, based on memory: but I’m inclined to go back and revise judgement at least on the set theory chapter.]
Summary verdict Now only for very selective dipping into by already-well-informed enthusiasts.
I was recently thumbing through Mathematical Logic (1967) by J.R. Shoenfield for a presentation concerning formal systems and it struck me out the peculiarity of his definition about a formal system on p.4. It is a standard one, in that he considers axioms, rules of inference and theorems as essential ingredients of a formal system. However, when he characterizes the theorems of a formal system he puts forward two conditions:
1. The axioms of F are theorems of F.
2. If all the hypothesis of a rule of F are theorems of F, then the conclusion of the rule is a theorem of F.
The second condition is just derivability from axioms to theorems through inference rules, but the first condition is awkward and puzzling, to say the least. Are the axioms of F derivable theorems of F? The only way that makes sense to me to consider axioms as theorems of F is by stipulating that axioms are theorems in which there are 0-use of inference rules. But, that is just a tough way to say that axioms are- formalized- suitable hypothesis about the formal system.
“The only way that makes sense to me to consider axioms as theorems of F is by stipulating that axioms are theorems in which there are 0-use of inference rules.” Which is fine by me. And is a useful convention to adopt since we can then neatly say that (in first-order logic, say) a wff is a theorem if an only if it is a logical truth (true in all models). Otherwise we’d have to be untidily disjunctive and say that a logical truth will be either an axiom or a theorem. And have to say that different axiomatisations of first order logic have different theorems, rather than regard different logical systems as providing different axiomatisations of the same theorems. And so on.
Not sure if it is on the list of Big books, and especially for philosophers:
Curry “Foundations of Mathematical logic” not so much about godel and so , but very mutch about what is an implication, a negation ect.
With all the book’s limitations from our current point of view, it should be remembered that it was immensely influential in its days. Generations of logicians cut their teeth on the book. And it was said that Shoenfield’s book marked the last time that the entire field of mathematical logic could be surveyed in one book. The old clichè is right — we are standing on the shoulders of giants.
I agree on both counts, Aldo. Perhaps I’m being grouchy and have overdone the negativity: but on dipping back into the book the other day (after a *long* time) memories of its being strikingly unapproachable and user-unfriendly seemed to be confirmed.
Since this book is often mentioned and recommended, it’s good to know of it’s limitations for someone trying top get into the subject; so this will be a very useful entry.
However, Shoenfield’s justification of the set theory axioms in this book and in a chapter of the Handbook of Mathematical logic is discussed in other books such as Hallett’s on Limitation of Size, so for that (and other reasons) the set theory part may still be of interest.
Re “other reasons”, I think forcing is still what Timothy Chow called it, an “open exposition problem”, making it more than usually useful to read different accounts, and I think Shoenfield’s is one that’s still worth reading.
(BTW, it seems to be “Shoen …” rather than “Schoen …”. Not that I knew. I only noticed when trying to look some things up online.)
Many thanks for this. “Shoen …” indeed: I’ve corrected what my unruly fingers typed contrary to my brain’s directions. And I guess I should take another look at the set theory chapter with your remarks in mind before writing up the Guide.