TITLE: What does Paul Halmos mean here?
QUESTION [6 upvotes]: 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 $(*)$.

REPLY [2 votes]: 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.