12
$\begingroup$

I have been spending the last month or so reading and doing the exercises of Chapters 1-6 of Jech's text, however I noticed a pattern in how Part I is constructed. The way it looks, of the 12 chapters of Part I, the first 6 chapters are on pretty basic set theory that any first course would cover (ZF axioms, ordinals, cardinals, choice, regularity), and the second 6 chapters seem to be applications of these ideas to other fields of mathematics. For instance, a chapter on combinatorics, a chapter on real analysis and measure, a chapter on Boolean algebra, etc. Some of this doesn't fit this pattern, but overall, it seems like chapters 7-12 are pretty skippable for forcing. So I go ahead and start Chapter 13.

Once I turn to Chapter 13, I get lost very quickly. It looks like this part of the text is much more logic-heavy than the other parts were. I have a good understanding of propositional and predicate logic, but no background in model theory or anything else. So my question is as follows:

For those who have Jech's text: are chapters 7-12 really skippable to start forcing? And what is the recommended background in logic to start forcing? Say... chapters 13-15 in Jech.

For those who don't have Jech's text, as I want to make this question more accessible: Forget about the chapter numbers and the specific text. How much logic should I know and review before starting forcing?

$\endgroup$
2
  • 1
    $\begingroup$ Chapters 7-12, far from being „applications of these ideas to other fields of mathematics“, are introductions to various highly central and important themes in set theory. This material is essential to anyone wanting to get a good idea of the subject overall. You might also find those chapters more readily accessible than later ones (such as forcing), and should serve to prepare you more intuitively (and in fact directly, in particular chapters 7 and 12) for it. So my advice would be just to keep going linearly through. If you really want to skip parts, read at least chapters 7 and 12 first. $\endgroup$
    – Farmer S
    Jul 5, 2021 at 11:22
  • $\begingroup$ Re chapter 13: It’s probably also pedagogically better to read chapter 13 before chapter 14, and there is some direct dependence, but the development of the main theory of forcing doesn‘t rely on chapter 13. $\endgroup$
    – Farmer S
    Jul 5, 2021 at 11:32

1 Answer 1

8
$\begingroup$

The fastest route to forcing is probably Nik Weaver's Forcing for Mathematicians. It gets you from the definition of an ordinal to the independence of $\sf CH$ in about 50 pages.

I would suggest using a different text than Jech for forcing. He uses Boolean algebras, which are sort of deprecated nowadays (Jech himself sort of abandons them rather quickly), and his treatment is very terse. Kunen's Set Theory would be a better introduction. I think the preface (of the 2013 edition) tells you exactly which chapters you need to read to understand the chapter on forcing.

There's a very new (2021) book by Mirna Džamonja called Fast Track to Forcing, whose title seems very relevant, but I haven't read it and thus can't comment on it.


To actually answer your question, you can happily skip chapters 7-11 in Jech (this refers to the 3rd edition). Chapter 12 is mandatory reading though, since that's where absoluteness is introduced.

For uses of forcing which require large cardinals or finer combinatorics, you might need to go back and read some of the stuff you skipped, but I think the above is sufficient to understand the basics. Still, I would once again encourage you to look at Kunen instead. If you must read Jech, having a copy of Bell's Set Theory: Boolean-Valued Models and Independence Proofs might be a good idea, as he gives a lot more details than Jech does.

$\endgroup$
8
  • $\begingroup$ This is really helpful information. A few things. Firstly, could you explain a little bit more on what you mean by Jech's treatment of forcing being terse? Secondly, do you think his chapters on Large Cardinals and Forcing, i.e. Ch. 16-21 are well written? I'd want to read more of it. I've not had problems with Jech thus far, and I suppose the only reason I'm sticking to it is because it's nice to have everything set theory related in one text. It's rather encyclopedic. $\endgroup$
    – Luna145
    Jul 5, 2021 at 6:13
  • $\begingroup$ I wouldn't recommend Mirna's book as an introduction to forcing for people who don't know enough set theory to begin with. It's a nice book with plenty of examples, but I don't know if the question "What logic do I need before approaching forcing" is the one that the book answers. $\endgroup$
    – Asaf Karagila
    Jul 5, 2021 at 8:03
  • $\begingroup$ It’s not true that boolean algebras (in connection with forcing) are „depreciated“ (deprecated?). Boolean algebras are very important, useful, and commonly feature in forcing arguments. In terms of introducing forcing, it is just a slightly different way to do it. I don’t have much idea how many people tend to use/prefer one method or the other for learning it. But I also tend to prefer using the approach in Kunen, as it seems to me a bit more concrete. $\endgroup$
    – Farmer S
    Jul 5, 2021 at 11:58
  • $\begingroup$ @FarmerS Yeah I may have been too harsh with my phrasing (and thanks for spotting the typo!). I do think the poset approach is easier to grasp (with all the details included) for someone who's just getting started. $\endgroup$
    – Reveillark
    Jul 5, 2021 at 15:57
  • $\begingroup$ @Luna145 All I mean by "terse" is that a lot of details are missing from the arguments, to the point that someone who's not familiar with the material might struggle to fill in the gaps. I agree that Jech is encyclopedic, and some of the chapters are great introductions to topics in Set Theory. I just don't think the chapter on Forcing falls into that category. $\endgroup$
    – Reveillark
    Jul 5, 2021 at 15:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.