Which is the best book on axiomatic set theory? I am interested in a book that is suitable for graduate studies and it is very mathematically rigorous.
[Math] Best book on axiomatic set theory.
book-recommendationreference-requestset-theory
Related Solutions
The title of Halmos's book is a bit misleading. He goes through developing basic axiomatic set theory but in a naive way. There are no contradictions in his book, and depending on your background that may be a good place to start. Halmos will still develop all the axioms of ZFC in his book, but they will be presented in natural language and a much slower pace than most axiomatic set theory books. If you are looking for something a bit more advanced, I would recommend either Set Theory by Ken Kunen or Set Theory by Thomas Jech.
The other thing is that set theory has a close relationship with mathematical logic, and so to understand the basics of set theory there is usually an assumed knowledge of some basic mathematical logic. For a mathematical logic book, I would recommend Mathematical Logic by Ebbinghaus and Flum or Introduction to Mathematical Logic by Enderton.
Either way, I think Naive Set Theory by Halmos should be a good beginning point. It is much shorter than the other books and does not require as much in the beginning.
Now, I have been reading the book. I summarize
Axiom 1 (of Separation). Let $A$ a set, and for each $x \in A$, let $\varphi(x)$ a property pertaining to $x$. Then there exists a set $C := \{x \in A : \varphi(x) \text{ is true} \}$ (or $\{x \in A : \varphi(x)\}$ for short), whose elements are precisely the elements $x$ in $A$ for which $\varphi(x)$ is true. $$\exists C \, \forall x \; (x \in C \iff x \in A \;\land\; \varphi(x) ).$$
Axiom 2 (of Extensionality). Two sets $A$ and $B$ are equal, $A = B$, if every element $x$ of $A$ belongs also to $B$, and every element $y$ of $B$ belongs also to $A$. $$\forall x \; (x \in A \iff x \in B) \implies A = B.$$
Your theorem
Theorem 3. Let $A$ and $B$ be sets. Then exists a unique set $C$ whose elements belongs to both $A$ and $B$. $$\exists ! C \, \forall x \; (x \in C \iff x \in A \;\land\; x \in B).$$
Now, to prove the theorem we need show the existence and uniqueness. First the existence. Let $A$ and $B$ be sets, and let $\varphi(x) := x \in B$. Then, by Axiom 1, we have $$\exists C \, \forall x \; (x \in C \iff x \in A \;\land\; \varphi(x) ),$$ i.e., $$\exists C \, \forall x \; (x \in C \iff x \in A \;\land\; x \in B ).$$ This prove the existence of the intersection set $C$ for any sets $A, B$.
We now show the uniqueness. Let $A$ and $B$ sets. Suppose there exists two sets $C, C'$ such that $$\exists C \, \forall x \; (x \in C \iff x \in A \;\land\; x \in B )$$ and $$\exists C' \, \forall x \; (x \in C' \iff x \in A \;\land\; x \in B ).$$ Using the notation, we have $$x \in A \;\land\; x \in B \iff x \in C',$$ i.e., $$x\in C \iff x \in A \;\land\; x \in B \iff x \in C'.$$ This means, the statements $x\in C$, $x \in A \;\land\; x \in B$, and $x \in C'$ are equivalents. Thus, we have $$x \in C \iff x \in C'.$$ By Axiom 2, we have $C = C'$ as desired. $\;\Box$
Now, note that to use the Axiom 1, you need two sets: a reference set from which construct the intersection, and another to define the property. Also, another valid definition of intersection is $$A \cap B = \{x \in A : x \in B\}.$$ So the set $z$ that you mention is $A$, and your property $x \in A \; \land \; x \in B$ should be just $x \in B$.
Finally, in Set theory, the Axiom of union exists because this does not follow from the other axioms (empty set, extensionaity, separation, etc). These axioms allow us to build smaller sets from other reference sets. But the union set is a larger set from anothers, because of that this axiom is necessary.
Best Answer
I've found Kunen's book "Set Theory: an Introduction to Independence Proofs" to be very good. I've heard Jech's book is good also.