# A nice introduction to forcing

I want to get acquainted with forcing, along with a few friends, and I'm looking for a text to introduce the basic notions (pardon the pun :) ).

The point is to study a text (or texts, if they can be reasonably seamed together) together for about one week, several hours a day, each of us (three) preparing a part on his own so as to explain it to the others (and so understand it better himself).

Again, I'm not looking for something necessarily in-depth, just enough to understand the basics at a decent, but not overly exerting pace (after all, it's still summer vacation for us :) ).

It is important to mention that all of us are somewhat familiar with the basics of descriptive set theory and model theory (but not axiomatic set theory per se), so ideally I would like a source that would somehow capitalize on that familiarity.

Any suggestions?

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What are you trying to do? Are you trying to simply get an understanding of how forcing works, and the basic mechanics of how it works? Or do you have some more ambitious goal like understanding the proof of the independence of CH (for example)? – Asaf Karagila Sep 8 '12 at 14:13
Try the new book by Kunen : Set Theory. – William Sep 8 '12 at 17:16
@AsafKaragila: I don't plan to study anything in particular just yet, rather, I want to get to know the language, so that I can more easily understand set theory related courses and seminars I hope to attend in the following years. To put it simply, I mostly want to cure my almost utter illiteracy in the subject. :) – tomasz Sep 8 '12 at 17:45

In addition to Asaf's recommendation above, I'll put a word in for Halbeisen's Combinatorial Set Theory: With a Gentle Introduction to Forcing. The book is divided into three parts. The first gives a basic introduction to axiomatic set theory along with some basic notions from combinatorial set theory (so as to have some goals for the next two parts). The second part contains a reasonable introduction to forcing, by first going through Martin's Axiom, to get the reader used to the idea of generic filters, and then proceeds to forcing proper (including product and iterated forcing; it does not, however, mention proper forcing). The last part contains a fairly thorough look at several important forcing notions, such as Cohen forcing, Miller forcing and Sacks forcing.

On the down-side, I cannot say that a previous knowledge of model theory or descriptive set theory would be incredibly helpful in going through this text (other than that knowledge of some model theory is in general useful when forcing).

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I think this probably contains all I wish to know, though I'll probably need to spend some time selecting the material. :) I think it's important to mention that the book is available for free on the author's website: user.math.uzh.ch/halbeisen/publications/publications.html . – tomasz Sep 9 '12 at 0:44

You can try A beginner's guide to forcing by Timothy Chow.

In term of books you can try Jech's small book about the axiom of choice. It has a nice half a chapter definition of forcing. It works with Boolean valued models, though.

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The short chapter on forcing in Boolos & Jeffrey's Computability and Logic will not bring you to the point of having sufficient proficiency to do anything, but I think there's still something to be said for it. B&J's book is intended to show a mathematician with no acquaintance with the subject that there are a number of interesting things in it. The book goes for breadth rather than depth and treats many topics, each very briefly but rigorously. The chapter treats forcing in arithmetic, not forcing in set theory. It gives a definition, proves a few basic properties, then shows you that forcing can be used to prove one substantial theorem. If you want to develop skill in the use of the technique in a way that involves doing lots of exercises, or if you want to learn how to use it in set theory, you'll need other sources. But this may be a good way for a mathematician to find out for the first time that such a technique exists.

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If one wants to see a forcing proof that isn't done in ZFC, Dana Scott has a nice exposition of the proof of the independence of CH: springerlink.com/content/hh339022jt1m5183 – Michael Greinecker Sep 10 '12 at 9:40
Or you can read about forcing in the context of computability theory and reverse mathematics. For example, in The Structure of Models of Peano Arithmetic by Roman Kossak and Jim Schmerl. The literature in the context of reverse mathematics grows every day. – Andrés E. Caicedo Jul 27 '13 at 18:36