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In the book "Elements of Set Theory" by Enderton, the union of an infinite collection of sets is described this way on pg.23:

Suppose we have the infinite collection of sets $A=\{b_0,b_1,b_2,\dots\}$, and we want to take the union of all $b_i$. Then we have a define a new axiom: the Union Axiom. It states that for any set $A$, there exists a set $B$ such that $$\forall x[x\in B\iff (\exists b\in A)x\in b]$$

Before this, regarding unions, we had only the Pairing Axiom, which says that for any two sets $u,v$, there exists the set $\{u,v\}$. This can then lead to: for any two sets $\{w\},\{u,v\}$,we have the set $\{u,v,w\}$, and so on. Hence, we can build finite sets using the Pairing Axiom.

My question is: How could $A$, the infinite collection of sets, ever be built using just the pairing axiom? And if it can never be built, how can it be the basis for a new axiom (namely the Union Axiom).

Thanks in advance!

algebraically_speaking
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  • The Seventh ZF axiom ("The Axiom of Infinity") posits the existence of an infinite set. This is an axiom. – Frank Mar 21 '14 at 02:36
  • @Frank- Thanks! I haven't yet come across this axiom. The book is inconsistent in itss development of theory, I suppose. – algebraically_speaking Mar 21 '14 at 02:41
  • No problem :) Logically it isn't inconsistent (by the sounds of it), because it just says if there is an infinite set of sets $A$, then... It doesn't actually claim that there does exist such an infinite set $A$. Though without looking at the book I can't say with certainty! – Frank Mar 21 '14 at 02:48
  • Though you are right, it is strange to state the Union Axiom in terms of infinite sets. Normally the Union Axiom is stated in terms of the 'union' of just two sets (as far as I remember!). – Frank Mar 21 '14 at 02:49
  • Given two sets ${u,v}$ and ${w}$, the Pairing Axiom leads to the two-element set ${{u,v},{w}}$ containing the two elements ${u,v}$ and ${w}$. From there, an application of the Union Axiom will get you the three-element set ${u,v,w}$. – bof Mar 21 '14 at 03:01
  • Yes, having double checked I apparently do remember wrong :) – Frank Mar 21 '14 at 03:06
  • The union axiom is not stated in terms of infinite sets, the motivation was; the axiom itself does not discuss finiteness or not of $A$. Anyway, though I do not know whether Enderton was aware of it, there is a good reason for motivating the axiom in terms of infinite unions: In the presence of the other $\mathsf{ZFC}$ axioms, the existence of unions of finitely many sets can be proved. See here. – Andrés E. Caicedo Mar 21 '14 at 06:03

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As Andres Caicedo put it in the comments:

The union axiom is not stated in terms of infinite sets, the motivation was; the axiom itself does not discuss finiteness or not of A. Anyway, though I do not know whether Enderton was aware of it, there is a good reason for motivating the axiom in terms of infinite unions: In the presence of the other ZFC axioms, the existence of unions of finitely many sets can be proved. See here.

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