You need something good to read over the holidays; so here is the Teach Yourself Logic 2016: A Study Guide!
For anyone who doesn’t know TYL, it is aimed at philosophers (who have already done a baby logic course) and mathematicians who are trying to teach themselves some mathematical logic and need a guide through the very large literature.
For those who are familiar with the 2015 version, there are some presentational changes aimed at making the Guide look a bit less daunting and more manageable, a few new recommendations or changes of view, and a lot of minor tinkering.
As always, constructive comments are immensely welcome. And many thanks to those who have made suggestions over the last year. I fear I may have overlooked/forgotten about some good ones: in which case, nag me! But most logic teachers are busy people, and haven’t time to wade through an 89 page Guide, however zestfully written it is. So, in the new year, I might start putting some cut-down excerpts here on the blog, to ask for feedback on specific sections. I’m sure all kinds of improvements could be made; but having spent quite a bit of time on the new version already, and to prevent myself tinkering away more over the holidays when I should be doing more fun things, here it is. Enjoy!
Since I seem to be the main person who suggested George Tourlakis’s Set theory, I’m going to try to defend it. I agree with your comments, up to a point, but I think you end up giving a misleading impression of the book.
Your points against the book are:
* Excessive and unnecessary formalism.
* Simple constructions and results take a long time to arrive.
* The final chapter on forcing is less clear and should be omitted at this stage.
I will take them in that order, but first a more general comment. I don’t think Tourlakis is the best choice as a first set theory book, but I would also say that about some of the other books listed in the same section. Instead, its role is more as a supplement or a second book, to overlap, go further, and take a different approach (as in the strategy described in section 1.3 of the Guide). I think its coverage of relatively advanced topics is what makes it interesting and useful. In your description, it’s not even clear that it covers them. (You do mention the chapter on forcing, but a chapter on forcing could just be a high-level overview.) So:
1. Excessive and unnecessary formalism.
Compare what Tourlakis says about confinality with, for example, Devlin. I don’t see any excessive formalism in Tourlakis there. I like Devlin’s book too, but Devlin’s more formal here. Devlin gives a formal definition and then starts using it. Unlike in Tourlakis, there’s little, if any, attempt to motivate the concept or to give the reader an intuitive feel for it. Most books are like Devlin’s when it gets to such topics. (Enderton is an exception, but he just barely gets to cofinality at the end.)
It’s true that Tourlakis uses formal proofs more than other books — I think that’s all the ‘excessive formalism’ amounts to — but that’s in the early parts of the book. He moves progressively to informal proofs. Most books expect readers to take it on faith that proofs can be formalised. Tourlakis doesn’t, and I think that pays off, to an extent, at some points later on; but I agree many could find it off-putting. If Tourlakis is their 2nd set theory book, however, they can go quickly through those parts.
I think your example is atypical and misleading. Most proofs in the book aren’t that formal; most aren’t about something so trivial. That proof appears when Tourlakis has introduced powersets, reminds us that something that seems obvious might not be, and so provides a formal proof that in a simple case no subset is left out. The proof plays a very minor role in the book. Also, while it does take a page, there is a lot of white space. It alternates lines of formal proof with brief exposition. The formal lines are short and there is a blank line before and after each one.
2. Simple constructions and results take a long time to arrive.
Well, some do. However, (1) Tourlakis gets to some topics that many books don’t reach at all, and (2) the primary reason for the late arrivals is the topic-order Tourlakis has chosen. (Someone reading the Guide might be forgiven for thinking it’s because of horrendously excessive formalism.)
3. The final chapter on forcing is less clear and should be omitted at this stage.
I agree that it should be omitted at this stage (Beginning set theory); but (unlike Devlin’s) the book isn’t mentioned later.
As for being less clear, I think that’s largely because it’s about forcing. It’s not for nothing that forcing is an open exposition problem while other topics aren’t. I would argue that Tourlakis gives a better and more complete account of the sort of forcing he discusses than Devlin does of the boolean-valued approach. I also think the Tourlakis forcing chapter is clearer when read after earlier chapters in the book than it would be on its own.
Two things on forcing:
The first mention of the boolean-valued models version of forcing is re Devlin’s Joy of Sets in 7.1.5, “Forcing further explored”. It might be worth mentioning that that approach is also explained (and used) in Jech, Set Theory.
There are a couple of things at about the same level as Chow’s ‘A beginners guide to forcing’ that also have similar aims. They are Chow’s earlier ‘Forcing for dummies’ and Kenny Easwaran’s ‘A cheerful introduction to forcing and the continuum hypothesis’. They are available online:
http://www-math.mit.edu/~tchow/mathstuff/forcingdum — for dummies
http://arxiv.org/pdf/0712.2279.pdf — cheerful intro
(Chow’s ‘beginners guide’ uses boolean-valued models; ‘Forcing for dummies’ doesn’t and so remains useful. Also, ‘for dummies’ is in plain text.)
One more: Robert Wolf’s A Tour Through Mathematical Logic also covers forcing at an approachable (though more elementary) level. In ‘beginners guide’, Timothy Chow says ‘Easwaran and Wolf give very nice overviews of forcing written in the same spirit as the present paper’. I think Wolf’s Tour may have been mentioned in an earlier version of the Guide.
Many thanks for these [and apologies that for some reason they didn’t automatically post straight away, as comments from known people usually do].