I'm learning ZFC set theory and I'm very confused about the Axiom of Pairing.
Axiom of Pairing: For any $a$ and $b$ there exists a set $\{a, b\}$ that contains exactly $a$ and $b$.
It seems that this axiom can be derived from other axioms in a lot of ways. For example, consider the powerset of the powerset of $\varnothing$, which is $\{\varnothing,\{\varnothing\}\}$, and then we apply the Axiom Schema of Replacement we can get $\{a,b\}$.
I looked for Wiki. It says the "non-independence" as well.
I'm confused why the Axiom of Pairing is important and independent. Also, why do we use "pair sets" instead of "singleton sets"? It seems equivalant that we use "singleton sets" axiom and the Axiom of Union to deduce the Axiom of Pairing?
Best Answer
Note that Replacement cannot be philosophically justified non-circularly, so philosophically you would want to distinguish between theorems that rely on Replacement and theorems that do not. Pairing is in contrast rather innocuous, and is crucial in obtaining ordered pairs (meaning definable pairing and projection) in Zermelo set theory.
Also, even if one axiom implies another, it does not imply that the weaker one is unimportant. For example, the unrestricted comprehension axiom (schema) of naive set theory entails Pairing and Union and Powerset and Replacement, but also entails contradiction. What is important is what can be justified and not usually what can imply more things.
In the case of Pairing, it captures a relatively simple notion that given any two objects $a,b$ we can form a single collection whose members must each be either equal to $a$ or $b$. 'Clearly' we can form such a conceptual collection.
It is true that if ZFC is meaningful then Replacement can be considered more fundamental than Separation and Pairing, because it not only entails them but also itself corresponds to reification of definable functions on a set, meaning that for any formula $P$ such that $\forall x \in S\ \exists! y ( P(x,y) )$ ($P$ is a definable function) there is an object $f$ representing $P$, namely $\forall x \in S\ \exists! y ( (x,y) \in f )$. Note that unrestricted reification of definable functions on a set entails a contradiction, just as unrestricted reification of definable 1-place predicates (which is precisely unrestricted comprehension) entails a contradiction.
But to give an example showing that it is not philosophically sufficient to have mere consistency, consider that PA+¬Con(PA) proves everything that PA does, and is consistent, but most logicians reject it as meaningless to the real world, unlike PA+Con(PA), which most logicians accept as true (or a good approximation) of real-world 'natural numbers'.