Lately I've been studying (on Jech "Set Theory" and Kanamori "The Higher Infinite") some problems related to the extension of measures.

I've never studied set theory before and I'm quite baffled from these results such as “if there is a such and such measure, then there exists a weakly inaccessible cardinal smaller than $2^{\aleph_0}$” and so on.

My interest is not directly in consistency problems, but I can't ignore them completely and I really don't know much about this field (I discovered only yesterday about Cohen proving the independence of the continuum hypothesis).

Would someone be kind enough to summarize the main results and/or tell me where to look (books/articles) which might help me have a general idea about the modern results in this field?

Thank you in advance!


For the easier stuff Drake's book is really good; Kanamori and Jech (while both fantastic texts) cover such a lot of ground that one can feel left behind at times. The proofs are also left in relatively `intuitive' form, allowing one to get a feel for the structures in question:

The reference is:

Frank R. Drake, Set Theory: An Introduction to Large Cardinals, North-Holland, 1974

One should note that this does not represent the entire large cardinal discussion to date. However, it is a really good text to get one's hands on the basic concepts in question before exploring the issues more thoroughly.

One good way to draw connections between raw cardinal size and consistency results is that if a cardinal $|\alpha|$ is sufficiently large then for some much smaller cardinal $|\beta|$, $V_{\alpha}$ will contain somewhere in it a model of $ZFC + \exists |\beta|$, thus establishing the consistency of the existence of a $|\beta|$ relative to an $|\alpha|$.

[EDIT 18 Sept 2013] Answer edited to reflect fact that the large cardinal discussion has moved on greatly since the publication of Drake's book.

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    $\begingroup$ It should be pointed out that the theory of large cardinals have developed A LOT in the past nearly-40 years since Drake's book was written. $\endgroup$ – Asaf Karagila Sep 17 '13 at 17:58
  • $\begingroup$ I absolutely agree; I have edited the answer accordingly. However, given the sheer volume of (often very esoteric) mathematics out there that involves large cardinals, Drake's book represents a good way to feel one's way in. Then once one has a handle on the basic concepts, one can start to expand one's horizons. I do concede, however, that as ``a summary of the main results'' it no where near complete. Then again, I wouldn't know where to start with a summary of the main results; there are just too many-I certainly shan't pretend to understand them all! $\endgroup$ – Neil Barton Sep 17 '13 at 23:12
  • $\begingroup$ @AsafKaragila, where in your opinion is the best place to start learning about large cardinals? $\endgroup$ – goblin May 12 '14 at 13:25
  • $\begingroup$ @user18921: I suppose a mix of Jech and Kanamori's books. Having a world expert disposable for asking questions (e.g. if they are giving a class on large cardinals) is also a good thing. $\endgroup$ – Asaf Karagila May 23 '14 at 8:53

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