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Has anyone in this community been through this book in its entirety? I started it about 18 years ago when I was just beginning undergraduate mathematics study and couldn't make any headway past the first three chapters.

Now before you all start asserting that this is a much more advanced text than the undergraduate level, let me just say that even now, with the benefit and maturity of graduate studies I still have trouble getting past chapter 5 in this text. I realize that this is a concise, condensed presentation of results (hence the scarcity of examples) but my problem with this text is the complete lack of motivation for its results. Take for instance the Consistency Theorem and Herbrand's Theorem in chapter 4. I've tried hunting down the original works where these results first appeared, for more background and examples, but these works either not available or, when they are, they're too obscure to read.

Reverting to other, more elementary texts, is of no use either. They're either too elementary and deal almost exclusively with the basic predicate calculus (as if everyone needs a new text daily for THAT), or are equally inaccessible and by the time you master the author's favorite notation you're already lost in the abstraction.

Is there any hope for material that introduces the advanced results more gradually, through background and examples, and more importantly through motivating questions? Motivation would be especially helpful with the advanced results of model theory presented in Chapter 5.

marcus66502
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  • Possibly some of the things I say (and cite) in my answer to Where to begin with foundations of mathematics could be of use. I've had a copy of Schoenfield's book since the late 1970s, and at the time (I was an undergraduate) I assumed that one day I would eventually know most of the stuff in his book, even if I didn't specialize in logic, but at the time despite being rather advanced (took at least 2 graduate level math courses each semester I was an undergraduate), I simply did not realize how vast mathematics is (continued) – Dave L. Renfro Jul 15 '20 at 17:22
  • and how little even a professional mathematician typically knows about many areas of mathematics. Anyway, getting more specifically to Schoenfield's book, reviews of it that I've seen (published in journals) indicate that it's not very user friendly, especially for non-experts. Incidentally, I've seen Schoenfield (I was an undergraduate at a university about 16-20 km away from his university) once, at one of his department's "general audience" colloquiums. The speaker (from elsewhere) discussed teaching basic calculus using nonstandard analysis (continued) – Dave L. Renfro Jul 15 '20 at 17:37
  • (may have been one of the authors of this 1979 book, as I think this was in Fall 1979 or Spring 1980), and Schoenfield asked several obviously pointed questions about the appropriateness of such an approach (in which one hand waves a bit about infinitesimals). I believe I've heard from others that his logic book worked much better in his classes than by self-study. You may want to get a copy of Hodel's logic book. Hodel was a colleague of Schoenfield, and much of his book is intended to expand on Schoenfield's. – Dave L. Renfro Jul 15 '20 at 17:43
  • Maybe Volume 1 of Tourlaskis – Mauro ALLEGRANZA Jul 15 '20 at 18:43
  • @ Dave L. Renfro I'm quite well aware that Schoenfield's text is not user-friendly. This was the gist of my post. The difficulty of this text points to problems with motivating the subject, which it does a terrible job of. The first motivational problem I have is in grasping the usefulness of studying syntax for its own sake. Schoenfield comments that since mathematical concepts are abstract and difficult to grasp, it is useful to study the form of theorems and the interplay between form and meaning. After spending countless hours on first-order logic I have yet to grasp the interplay. – marcus66502 Jul 15 '20 at 18:57
  • @DaveL.Renfro Whatever your area of specialty is in mathematics, you want to be able to prove new theorems in it. First-order logic is about proving theorems about the theory, not theorems within it. I have never seen an example in, say, analysis, where someone has gone "well we have this theorem of Analysis, which has this form, call it A. And there's a theorem of logic that says if a statement of form A is true, then a statement of form B is true, therefore, voila, we have a new theorem, of form B". The question should be obvious: how does the study of syntax help us get new results? – marcus66502 Jul 15 '20 at 19:02
  • Well, there is the situation in which there are problems in analysis (and other mathematical areas) that have been proven independent of ZFC (and many others which might be, but are still under investigation; and also the "degree of independence of ZFC" such as this is also something one can investigate), such as the many classical problems in Sierpinski's 1934 book Hypothèse du Continu, (continued) – Dave L. Renfro Jul 15 '20 at 19:52
  • and to pursue these types of things one needs a solid grounding in logic and model theory at the level of Schoenfield's text (last third especially), something I don't have by the way. – Dave L. Renfro Jul 15 '20 at 19:54
  • I have checked out most recommended references. One issue I have is that there's no way to get to what I'm after, in these books, without going though the hassle of familiarizing myself with each author's notation, and that requires reading the text from page one -- not something I want to do (I don't need to spend time on introduction to first order syntax and theories; I have an understanding of the basics). There's no shortage of texts covering the basics, but I have yet to see something that fleshes out the results of model theory through motivating questions, and without heavy notation. – marcus66502 Jul 15 '20 at 20:17
  • I certainly sympathize with the notation hurdles each author seems to set up for the reader. Maybe A Tour Through Mathematical Logic by Robert S. Wolf, although for some reason the prices at amazon are astronomically high (I don't think I paid more than $40 or so for a new copy within the last 10 years). It's a lot cheaper through the MAA, but the quoted price might be for members. – Dave L. Renfro Jul 15 '20 at 21:14
  • Contents of Wolf's book. Reviews of Wolf's book: Bulletin of Symbolic Logic AND American Mathematical Monthly AND Monatshefte für Mathematik. – Dave L. Renfro Jul 15 '20 at 21:20
  • @marcus66502 Note that Hodel's book follows the same notation conventions as Shoenfield, so if this is your problem, I definitely recommend it. – Nagase Jul 15 '20 at 23:07
  • marcus66502 and @DaveL.Renfro Is Ebbinghaus' Mathematical Logic a good alternative to Shoenfield's? – Tim Jul 25 '20 at 20:22

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Shoenfield's book is rather notoriously tough going.

Over fifty years(!) after publication there are quite a few nicer alternatives. Hodel, as recommended by @Nagase is respectable but patchy: it would not be my a first choice -- see http://www.logicmatters.net/tyl/booknotes/hodel/

On the whole I'd recommend looking for accounts in different books of (i) FOL and elementary model theory, (ii) computability and formal arithmetic, and (iii) set theory. [Merely notational differences between texts shouldn't really give you pause: a quick skim is usually enough to pick up local idioms.]

You'll find a detailed 2020 guide to books suitable for self-study at various levels in those different areas linked here: https://www.logicmatters.net/tyl/ (last updated, as it happens, a few days ago). This Guide is long, but you should find (I hope!) enough signposts to the parts that are relevant to your interests/your mathematical level.

Peter Smith
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