# What does Paul Halmos mean here?

In Naive Set Theory, in Section 1.3 "Unordered Pairs", Paul Halmos mentions the following:

If, temporarily, we refer to the sentence $$”x=a \text{ or } x=b”$$ as $$S(x)$$, we may express the axiom of pairing by saying that there exists a set $$B$$ such that $$x\in B\text{ if and only if } S(x).\tag{*}$$ The axiom of specification applied to a set $$A$$ [such that $$a\in A\text{ and } b\in A$$, whose existence is guaranteed by axiom of pairing], asserts the existence of a set $$B$$ such that $$x\in B\text{ if and only if } (x\in A\text{ and } S(x)).\tag{**}$$ The relation between $$(*)$$ and $$(**)$$ typifies something that occurs quite frequently. All the remaining principles of set construction are pseudo-special cases of the axiom of specification in the sense in which $$(*)$$ is a pseudo-special case of $$(**)$$.

Question: What does Halmos mean by stating that remaining principles of set construction and $$(*)$$ are pseudo-special cases of axiom of specification and $$(**)$$, respectively?

In fact, $$(**)$$ seems a special case of $$(*)$$.

I think the general principle here is that if you want to show that there exists a set containing precisely those elements $$x$$ satisfying some first-order formula $$S(x)$$, it is sufficient to show there exists a set containing at least those elements, and then invoke the axiom of specification. This is what's done when we use (**) to establish (*) - the original axiom of pairing, as stated by Halmos, asserts there exists a set containing $$a$$ and $$b$$, but allows for the possibility that all such sets contain additional unwanted elements. You need specification to rule out the latter.

So whenever one writes something like "let $$A$$ be the set of all $$x$$ such that $$S(x)$$", there may be a hidden use of the axiom of specification.

Without Pairing, we cannot use Specification (a.k.a. Comprehension) to obtain any set with exactly 1 or 2 members. Because without Pairing, we cannot show that for any $$x,y$$ there exists $$A$$ with $$x\in A$$ and $$y\in A.$$

I have no comment on what "pseudo-special case" means.

The issue is with the so-called (unrestricted) Principle of comprehension :

$$\exists B \ [x \in B \leftrightarrow S(x)]$$.

If we have the said principle available, we can prove that, for $$a,b$$ whatever, the pair $$\{ a, b \}$$ exists, using it (as Halmos says) with $$x=a \lor x=b$$ as the formula $$S(x)$$.
But in axiomatic set theory, the above principle is replaced by Specification : $$x \in B \text { iff } (x \in A \text { and } S(x))$$.
Thus, in order to prove that the pair $$\{ a, b \}$$ exists, we have to find a previous existing set $$A$$ to which $$a$$ and $$b$$ belongs.