I am trying to understand section 3 of Naive Set Theory by Halmos.
First assume that there is a set, then by axiom of specification, empty set exists. And then he motivates the necessity of axiom of pairing by the following reasoning:
For all we know there is only one set and that one is empty. Are there enough sets to ensure that every set is an element of some set? Is it true that for any two sets there is a third one that they both belong to? What about three sets, or four, or any number? We need a new principle of set construction to resolve such questions. The following principle is a good beginning.
Axiom of pairing: For any two sets there exists a set that they both belong to.
My question is, how does Axiom of pairing help if we only have one set to begin with?