Prerequisites for Forcing

Question

What are the prerequisites to study the forcing method by Cohen? Which book has a modern notation on the subject? Thanks in advance.

Answer 1

I am a big fan of Halbeisen's "Combinatorial Set Theory: With a Gentle Introduction to Forcing".

The only issue there is that Halbeisen is forcing "up", that is, $p\leq q$ means that $q$ is a stronger condition. This is something a lot of people are moving away from these days (although not everyone). So this is just something to be aware of, when you start reading other people's forcing proofs.

Other than this, a quick bang for your buck would be Jech's "Axiom of Choice" book which has a speedy exposition on set theory (skip Chapter 4 if you're interested in forcing), although it mainly focused on methods related to the Axiom of Choice, and at the end of the day, most actual proofs "in the wild" do not use Boolean-valued style arguments. It can still give you a good basic idea on what is going on. His "Set Theory" book (3rd edition, of course) is a great book as well.

Answer 2

Kunen's book "Set Theory: An Introduction To Independence Proofs" is pretty much self contained and one of the standard references on the topic, depending on your background in set theory you can skip the first chapters. Strictly speaking the prerequisites for forcing are not much, but for it to make sense and be motivated you should be familiar with some set theory, as presented in Kunen's book for example

Answer 3

Not a book, but a good source nonetheless: the notes and exercises here, by Sherwood Hachtman and Spencer Unger. They in turn build off of notes and exercises by Justin Palumbo, which unfortunately seem to be unavailable - Justin's notes were what I learned from (with Kunen as a supplementary text).

The main distinguishing feature of these notes is how they use Martin's Axiom as a major prelude to forcing itself. I know several people who dislike this approach, but for me personally it was wonderful: basically, it splits the effort by first showing how generic filters can be used to solve problems beyond direct applications of the Baire category theorem, which makes forcing much less mysterious.