Second reading on set theory? Any recommendations?

I have in past six-ish months studied through the Herbert Enderton's Elements of set theory book.

Up to the point the book is great,I loved most parts of it and learned almost everything up to the section on cardinal arithmetic.

Next I plan on reading Jechs Introduction to set theory and/or Set theory for working mathematician.

What I have wondered about is,what do I do next?Where can I master cardinals and see examples of relatively simple applications of axiom of choice(most examples are inclusive of advanced mathematics and the point of axioms use is obscured)?

Topics I am mostly interested in are:

      -Applications of axiom of choice
      -Cardinals and ordinals
      -Material on countable and uncountable infinities
      -Perhaps a easy introduction to descriptive set theory
      -Morse-kelly and Bernays-Von Neumann set theory
      -Transfinite induction

So what do you guys recommend?Share your wisdom

Solution 1:

Intermediate level set theory texts: [1], [2], [4], [8], [10], [12].

More advanced than these, but possibly within your present reach are [6] and [11].

Descriptive set theory can be found in some of these (e.g. [6] and [12]), and more fully in [9] and [14].

A lot of useful material on cardinal and ordinal numbers is in [5] and [13].

[3] is a good elementary reference for topics that are often omitted in standard elementary texts (e.g. arithmetic operations on linear orderings) and for historical/bibliographical information.

Finally, Handbook of Set Theory [see also here] might have some useful articles, but at this point I have not had a chance to look at it very carefully.

[1] Krzysztof [Chris] Ciesielski, Set Theory for the Working Mathematician, London Mathematical Society Student Texts #39, Cambridge University Press, 1997, xi + 236 pages. table of contents

[2] Frank Robert Drake and Dasharath Singh, Intermediate Set Theory, John Wiley and Sons, 1996, x + 234 pages. table of contents book review

[3] Abraham Adolf [Adolph] Halevi Fraenkel, Abstract Set Theory, 3rd revised edition, Studies in Logic and the Foundations of Mathematics, 1966, viii + 297 pages. review of 1953 edition

[4] András Hajnal and Peter Hamburger, Set Theory, London Mathematical Society Student Texts #48, Cambridge University Press, 1999, viii + 316 pages.

[5] Michael Holz, Karsten Steffens, and Edmund [Edi] Weitz, Introduction to Cardinal Arithmetic, Birkhäuser Advanced Texts, Birkhäuser Verlag, 1999, viii + 304 pages. book review

[6] Thomas J. Jech, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, 2003, xiv + 769 pages. 2006 edition 2011 edition

[7] Winfried Just and Martin Weese, Discovering Modern Set Theory. I. The Basics, Graduate Studies in Mathematics #8, American Mathematical Society, 1996, xviii + 210 pages. list of topics

[8] Winfried Just and Martin Weese, Discovering Modern Set Theory. II. Set-Theoretic Tools for Every Mathematician, Graduate Studies in Mathematics #18, American Mathematical Society, 1997, xiv + 224 pages. list of topics

[9] Alexander Sotirios Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics #156, Springer-Verlag, 1995, xviii + 402 pages. corrections and updates (4 October 2013)

[10] Péter Komjáth and Vilmos Totik, Problems and Theorems in Classical Set Theory, Problem Books in Mathematics, Springer, 2006, xii + 514 pages. table of contents book review

[11] Herbert Kenneth Kunen, Set Theory. An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics #102, North-Holland, 1980, xvi + 313 pages. 2011 edition review of 1983 edition

[12] Azriel Levy, Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag, 1979, xiv + 391 pages. Reprinted by Dover Publications in 2002. review

[13] Waclaw Franciszek Sierpinski, Cardinal and Ordinal Numbers, 2nd edition revised, Monografie Matematyczne #34, PWN--Polish Scientific Publishers, 1965, 491 pages. review of 1958 edition

[14] Sashi Mohan Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics #180, Springer-Verlag, 1998, xvi + 261 pages.