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?
Best Answer
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.