Good introduction to cardinals?

Question

is there a good text book to cardinals?

I am more interested in how cardinal works than cardinality. Because it seems in undergrad they cut off at proof of $\mathbb{R}$ is uncountable and does not go at all into $\aleph$ numbers, $\omega$ numbers.

I haven't found a standard, reliable reference to cardinals. Any help is deeply appreciated.

Answer 1

At the most elementary level you might start with E. Kamke's Theory of Sets.

Note that $\aleph_1$ is defined to be the cardinality of the set of all countable ordinals, and $\aleph_2$ is defined to be the cardinality of the set of all ordinals of cardinality $\le \aleph_1$, and so on through all $\aleph$s of finite index.

And $\beth_0$ is the same as $\aleph_0$ and for larger ordinals $\kappa$, $\beth_{\kappa+1} = 2^{\beth_\kappa}$, and if $\kappa$ is a limit ordinal then $\beth_\kappa = \sup\limits_{\lambda<\kappa} \beth_\lambda$.

Answer 2

An introduction to the theory of cardinals is given in any set theory text. For decades, the standard introduction to basic set theory was Paul Halmos' Naive Set Theory , which is written in Halmos' concise, clear style with tons of problems to chew on. The book's recently been reissued in a nice inexpensive paperback, so that's really a good choice.

My favorite book on set theory is Notes On Set Theory by Yiannis N. Moschovakis. It's more expensive and a little more difficult then Halmos, but it's also much more comprehensive and readable. It relates set theory to actual mathematical constructions,particularly the number systems and topological spaces. This is the book I wish I'd had when I learned set theory and it'll be one of your favs,too. That's the one I'd use if I could.

Answer 3

I like Set Theory: An Introduction to Large Cardinals by Frank Drake. Is old by good. Discusses what happens without/with Choice and according to Andreas Blass:

... deserves your attention even if you're not particularly interested in large cardinals. Despite its subtitle, it contains very nice presentations of a lot of general set-theoretic background before getting to the large cardinals.