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Jech's book on Set Theory is very often recommended and it is indeed an amazing collection of results. But to me it feels a bit like a reference book instead of a studying book; it seems to me you would use Jech's book if you're already fluent in a lot of the topics but want to brush up on something.

For example, A. Miller's notes "Infinite Ramsey Theory" (1996) where he talks about Ellentuck's topology and proves the Galvin-Prikry theorem were to me far more accessible and easy to read than Jech's chapter on this topic.

I am now interested on reading on Levy Collapsing and Solovay's model but as Jech's book hasn't been giving me a good experience so far so I'm looking for some recommendations on books where I can find this that would be more accessible than Jech's book. I've been studying forcing from Kunen's book "Set Theory" and I find Kunen's writing very clear and enjoyable, it saddens me that I couldn't find anything by Kunen on this topic.

Thank you.

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    $\begingroup$ Try Kanamori's The higher infinite. You may also want to consider Bartoszyński-Judah's Set theory. On the structure of the real line. $\endgroup$ – Andrés E. Caicedo Nov 15 '16 at 1:00
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    $\begingroup$ Have you read Solovay's original paper? I thought it was very clearly written. Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, 2nd Ser., vol. 82, no. 1 (Jul. 1970), pp. 1-56. $\endgroup$ – Mitchell Spector Nov 15 '16 at 3:09
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    $\begingroup$ @Andrés: Definitely Kanamori. You should post this as an answer. $\endgroup$ – Asaf Karagila Nov 15 '16 at 3:43
  • $\begingroup$ @AndrésE.Caicedo Thanks so much! I found Kanamori's book to be wonderful. If you think it's appropriate I'll gladly accept this if you post it as an answer. While it was interesting to attempt to read the original paper, it was more convoluted than needed as Kanamori's version simplifies the proof a bit. Also Kanamori's book is pretty self contained and Solovay's paper isn't. $\endgroup$ – JKEG Nov 24 '16 at 18:26
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This is a beautiful and truly fundamental result, and so there are several good quality presentations.

Try

MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3,

or any of the newer editions (the 2003 second edition, or the 2009 paperback reprint).

An alternative is

MR1350295. Bartoszyński, Tomek; Judah, Haim. Set theory. On the structure of the real line. A K Peters, Ltd., Wellesley, MA, 1995. xii+546 pp. ISBN: 1-56881-044-X,

which also emphasizes results on ZFC and the projective hierarchy.

And, of course, at some point I highly recommend you study Solovay's original presentation,

MR0265151. Solovay, Robert M. A model of set-theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2) 92 1970 1–56.

It is a very early result in the history of forcing, so there are a few details that look like oddities nowadays. But it also has many nice and useful ideas that more than reward the required effort. (For instance, a lot of recent work on large cardinals and forcing absoluteness requires a good understanding of the details of this result.)

Enjoy!

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  • $\begingroup$ Thank you! Could you list some of these recent results on forcing and large cardinals that you mentioned? I enjoyed very much studying through Solovay's result and would like something to follow it up. Perhaps some more on random reals as well. Thanks again. $\endgroup$ – JKEG Jan 9 '17 at 18:41

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