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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.

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3 Answers 3

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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.

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    $\begingroup$ So if I understand correctly, all set-building axioms (pairing, union, ..., etc.) would have been a consequence of the axiom of specification had it not been paradoxical. So, their truth was supposed to be severally assumed axiomatically in order to avoid also assuming the truth of the paradoxical results? $\endgroup$
    – Camelot823
    Jun 21, 2023 at 23:23
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    $\begingroup$ @Camelot823 the gist is that if you don't allow unrestricted comprehension (which is necessary because it leads to contradictions), you need axioms asserting what kind of sets are allowed. $\endgroup$
    – Alex
    Jun 21, 2023 at 23:25
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    $\begingroup$ @Alex yes, thank you, that makes all of the sense now! $\endgroup$
    – Camelot823
    Jun 21, 2023 at 23:31
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There are many ways to write down the axioms of set theory (i.e., ZFC) that are essentially equivalent to each other.

The version of the pairing axiom you are talking about is weaker than the stronger version that additionally asserts that the pairing set contains no other sets. We generally wish to state the axioms in their weakest possible form.

In ZFC we don't allow unrestricted comprehension because it leads to contradictions like Russell's paradox. We allow the axiom schema of restricted comprehension, i.e., the axiom schema of specification, along with other axioms that assert what kind of sets are permitted to exist.

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    $\begingroup$ By "weakest" you mean excluding information that can be obtained as a consequence of what was already stated, is that correct? $\endgroup$
    – Camelot823
    Jun 21, 2023 at 23:33
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    $\begingroup$ @Camelot823 exactly. $\endgroup$
    – Alex
    Jun 21, 2023 at 23:33
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The Axiom of Pairing states that, for any two sets, there exists a set containing exactly those two sets. It doesn't uniquely identify or specify the set, only guarantees its existence.

The Axiom of Specification allows forming a set by selecting elements from an existing set that satisfy a given property.

While you can use Specification to describe a set containing two specified sets, Pairing is considered a more direct and foundational principle in set theory. It ensures the existence of pairs without introducing unnecessary details about their uniqueness.

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