What does Paul Halmos mean here?

Question

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 $(*)$.

Answer 1

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.

Answer 2

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.

Answer 3

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

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

As we know, this principle leads to Russell's Paradox.

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.

Pair axiom allows us to avoid this detour, licensing the existence of the pair without further conditions.

In this sense, it is a sort of "limited comprehension" principle : we can sey that it is a special case of Comprehension.

In this sense, IMO, Halmos calls Pair a "pseudo-special" case of Specification.