In Halmos' book on Naive Set Theory, the author elucidates the role of the axiom of specification in developing the realm of set theory. According to Halmos, the axiom of specification is useful for asserting the existences of sets given by other axioms. Examples he gives in the book have to do with the Axiom of Pairing. The axiom of pairing states that for any two given sets, there exists a set that both belong to. However, he says, the axiom of pairing doesn't assert the existence of the pair itself, it merely says that there exists a larger set in which both sets are a part of. It is the duty of the axiom of specification to assert then that there also exists a set that contains both of those sets, and nothing else. I don't understand this then: why not just state the axiom of pairing as "for any two given sets, there exists a set that both are a member of, and no other set is". I also don't completely understand the necessity of the axiom of pairing, isn't it possible to derive it as a conclusion of the axiom of specification and a few logical operators?
By simply specifying: $A = \{x: x = B$ or $x = C\}$. Doesn't this already define $A$ as the set that is the pairing of $B$ and $C$?
Thank you in advance.
Best Answer
Russell established that being allowed to define $A = \{x:P(x)\}$, where $P(x)$ is an arbitrary predicate, leads to contradictions ($A = \{x:x \not\in x\}$ being the classic example). Instead we want the existence of a set $S$ containing both $B$ and $C$. Then we can define $A = \{x\in S : x=B \text{ or }x=C\}$ using the Axiom of Specification.