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What axiom makes it possible to take the union or intersection of an infinite number of sets $A_1, A_2, \ldots,$ and get a resulting set $B$.

In probability I've calculated $P(\cup_{i=1}^{\infty} A_i)$ and $P(\cap_{i=1}^{\infty} A_i)$.

However how can I calculate the probability of the event $\cup_{i=1}^{\infty} A_i$ ? The event is never completely determined ? I mean we can keep taking the union of sets, but we will never be done and get a resulting set. There could always be some set $A_j$ correspoding to $j\in \mathbb N$ that makes the union of greater cardinality ?

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  • $\begingroup$ At least in the Wikipedia version the existence of the union (without any restriction to finite sets) is one of the axioms. $\endgroup$
    – celtschk
    Apr 5 '14 at 10:39
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    $\begingroup$ I just noted that I should explicitly note that the union, as defined in that axiom, is not a sequential application of two-set unions, but a single operation that takes a set of arbitrary many (possibly even uncountably many!) sets to take the union of, and gives the union of those sety. Your unions are just the countably infinite special case. Note also that I didn't see a corresponding direct axiom for the intersection. $\endgroup$
    – celtschk
    Apr 5 '14 at 10:49
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Referring to the Wikipedia link in the comments (It's been a while since I studied set theory), intersection should follow from 'Axiom schema of specification'.

For an arbitrary collection $\mathcal{C}$ of subsets of some set $A$, we have $$\cap \mathcal{C} = \{a \in A \ | \ \phi(a) \}$$

where $$\phi(a) = \forall B \ (B \in \mathcal{C} \implies a \in B)$$

These sort of constructions are not as ill behaved as you might think. Often they can be re-expressed in terms of being a 'minimal' something or a 'maximal' something else. I don't know how familiar you are with topology, but it houses a few really good examples.

There they define the interior of a set as the union of all open subsets (of which there may be uncountably many), but you can define that equivalently as the maximal open subset: the unique open subset with the property that it's a superset of every other open subset.

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  • $\begingroup$ Thank you @G.H. Faust for you inspiring post. So in general we should not think so much on these infinite unions/intersections ? I've never had set theory, even thus we use these infinite unions/intersections all the time ? $\endgroup$
    – Shuzheng
    Apr 5 '14 at 11:25
  • $\begingroup$ @user111854 Outside of set theory, you probably should just take for granted these things can be done, but it does make a lot of sense to think quite carefully about these when set theory is the thing you're actually studying. I think part of the issue here is that you're expecting infinity to conform to your intuition (e.g. that of an infinite union to be built up sequentially), but that's usually asking a bit to much. All that really matters in set theory is what follows from the axioms and what's inconsistent with them. $\endgroup$ Apr 5 '14 at 11:36
  • $\begingroup$ But doesn't that derivation mean the axiom of union is redundant because it can be derived from specification using the formula $\phi(a)=\exists B(B\in C \land a\in B)$? $\endgroup$
    – celtschk
    Apr 5 '14 at 11:39
  • $\begingroup$ @celtschk: No, you need some set to bound the union in order to apply separation axioms. So in the case this set is given, sure. But in general it might not exist otherwise. $\endgroup$
    – Asaf Karagila
    Apr 5 '14 at 11:43
  • $\begingroup$ @AsafKaragila: I see, thank you. $\endgroup$
    – celtschk
    Apr 5 '14 at 11:46

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