Best Books on Set Theory: Axioms & Theory

[Vincent Mazzo]

What are the best books on Set Theory?? I mean a book with all axioms and theory and dispenses other books.

 

[micromass]

"Introduction to set theory" by Hrbacek and Jech is the best introduction
"Set theory" by Jech is my favorite, but it's only good if you already know a bit set theory.
"Set theory: an introduction of independence proofs" by Kunen is also good, but it's quite difficult...

 

[discrete*]

I just purchased Stoll's Set Theory and Logic. I've started to go through it a bit, and am liking it so far. It seems like it would be great for a complete beginner, however I have experience in both set theory and logic.

 

[quantum123]

What is an initial segment?

 

[honestrosewater]

I love https://www.amazon.com/dp/0521479983/?tag=pfamazon01-20 might be enough to satisfy you there).

 

[JeremyHorne]

Besides Stoll, a real classic is Patrick Suppes, Axiomatic Set Theory. Also, try Schaum's Outline Series on Set Theory.

 

[quantum123]

I have Patrick Suppes, Axiomatic Set Theory, but :-
1) There is no proof for Zorn's lemma. They say , this is just a simple exercise to be done by the students (that is plain stupid, if I know how to prove Zorn's lemma, why should I buy the textbook? simple logic)
2) The book did not even define initial segment properly.

I also have the Schaum series book on Set Theory by Seymour Lipschutz. .
This is better. They define initial segment properly. Also, they have an almost complete proof of Zorn's lemma from the Well-Ordering Theorem, which they also proved from the Axiom of Choice, even though the proofs are a bit long and indirect.

 

[JeremyHorne]

Quantum123, you are aware, I am sure, that the Axiom of Choice has been found by Paul J. Cohen in 1966 to be independent of Z-F set theory. Also, I see out there claims that Zorn's Lemma is equivalent to the Axiom of Choice, as for example: http://sporadic.stanford.edu/bump/math161/project.pdf .Now, if you can find a proof from Z-F set theory, itself ... .

In addition, perhaps the following might help:
https://docs.google.com/viewer?url=...hortProofs/Zorn.pdf&embedded=true&chrome=true

and: http://www.uwec.edu/andersrn/SETSV.pdf

and: http://www.math.cornell.edu/~kbrown/6310/zorn.pdf

As to "initial segment" in Suppes, I see reference to "R-Segment" on page 77. I haven't used Suppes for 35 years, so to save my doing an intensive search, is this page the one to which you refer?

 

[JeremyHorne]

Just caught this:

Quantum123 asks in this forum Old Dec25-10, 04:37 PM , "What is an initial segment? " and then says Unread T, 06:16 AM referring to to my post and Suppes, "The book did not even define initial segment properly." If quantum123 does not know what an "initial segment" is, how can s/he know whether Suppes (or anyone else, for that matter) is defining it correctly? Am I missing something here?

 

[quantum123]

I meant that I did not understand Patrick Suppes' definition but understood Seymour Lipschutz's definition, which I have not read at the time of posting Dec25-10, 04:37 PM .

 

[StatOnTheSide]

It is interesting that nobody mentions Enderton here. I would love to know your opinion on Enderton's book titled Elements of Set Theory. UCB seems to use it as a text for their Set Theory course.

 

[quantum123]

It is indeed a good book. Zorn's Lemma is proven in less than a page using ordinals and Hartog's theorem and Well-Ordering theorem at pg 198.
Thanks for the reminder. It is a good book for learning set theory.
But I think Goldrei's book is the best. Classic Set Theory.

 

[quantum123]

@StatOnTheSide:
Thanks to your recommendation, I have been reading Enderton's book on set theory. This book is the purest ZFC book I ever seen. Thanks.

 

[dustbin]

"Set theory" by Jech is my favorite, but it's only good if you already know a bit set theory.

How much would you consider adequate? I am familiar with sets from Apostol, Principles of Mathematics, and a couple of proof-intro books.

 

[micromass]

How much would you consider adequate? I am familiar with sets from Apostol, Principles of Mathematics, and a couple of proof-intro books.

That's not enough. You need to be familiar with ZFC axioms already and with the basic results. Try going through a book such as Hrbacek & Jech first.

 

[JeremyHorne]

There is always the old conundrum of whether logic "came before" mathematics of vice versa. I know I express my prejudice in asserting the first and surely agree with the micromass that one has to be knowledgeable w/ ZFC axioms before attacking math.

As to the "best" work on set theory? Surely one should read Z-F in the original, as well as Von Neuman, Cantor, and the other "prime sources" for set theories, but, ultimately is a knowledge of all of them that is the best, as each will have a different slant and aspect that only can rich the overall knowledge of set theory, itself. I also would throw into the pile "Logic for mathematicians" by J.B. Rosser. Discrete, I think, makes an accurate observation that Stoll is a good approach for a beginner. Also, the Schaum's Outline gives an excellent "repair manual" approach, with step-by-step worked exercises after an overview of theory.