A nice introduction to forcing

Solution 1:

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).

Solution 2:

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.