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Axiom of Pairing. Given two sets $C$ and $D$, there is one set $X$ whose members are exactly $C$ and $D$. In symbols (which I hate them):

$$ \forall C,D\; \exists X : \forall x\, (x\in X \Longleftrightarrow x=C \mbox{ or } x=D). $$

If Wiki is right, this Axiom can be deduced either from the axiom of empty set and the axiom of power set or from the axiom of infinity. So in some sense, my question is stupid.

Anyway, is the following statement equivalent to the Axiom of pairing?:

My Statement. If $X$ and $Y$ are sets, then $X\cup Y$ is also a set.

Obviusly, it is not my idea, it is said by the book I'm following. The set of axioms the author uses is very similar to MK I think, but he considers this new axiom instead of the usual Axiom of Pairing and he doesn't consider the Axiom of Empty set (but he considers the Axiom of Infinite, so it doesn't matter).

Thanks.

Dog_69
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  • Do you mean equivalent over the rest of ZF? – Leo163 Jan 30 '18 at 23:44
  • What is exactly said by the book you're following? Does it assert that the axion of union is equivalent to the axiom of pairing? – la flaca Jan 30 '18 at 23:51
  • Yes. Over complete theories. I don't know if over ZF or MK or NBG, because as the authors says ''Esentially, the axiomatic that we are going to use is the ZF, with substantial contributions of the system developed by NBG (...) and with some tweaks given by J. L. Kelly''. – Dog_69 Jan 30 '18 at 23:51
  • I think yoy won't know the book. It is Estructuras de álgebra multilineal of Joaquín Olivert, a teacher from univerity in Spain. And no. It uses the axiom of union and ''my axiom'', which I think it should be equivalent to the Axiom of Pairing. – Dog_69 Jan 30 '18 at 23:53
  • I am confused by the Wiki entry. How does the pair axiom follow from replacement ? Granted you have at least one two element set. Replacement is a schema, I can see how to produce a pair set schema, but to prove it with a universal quantifiers, I dont see. – Rene Schipperus Jan 31 '18 at 00:03
  • @ReneSchipperus: There is a question in Math Stack about that: https://math.stackexchange.com/questions/141646/proving-the-pairing-axiom-from-the-rest-of-zf I hope this helps you and then you can help me. – Dog_69 Jan 31 '18 at 00:10
  • Yeah, ok I see how to do it now. – Rene Schipperus Jan 31 '18 at 00:26

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My book follows EXACTLY (with the same names) the axioms in Kelly's General toplogy (see here). Moreover, Wiki says ''Kelly introduced his axioms gradually, as needed to develop the topics listed after each instance of Develop below'', and the same happens with the author of my book. So probably my next question will be about Kelly's General Topology...

Dog_69
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