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I'm reading Halmos's Naive Set Theory, and right now I'm on the section about the axiom of unions. As stated in the book, the axiom reads:

For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.

Essentially, U={x: x∈X for some X in C}. My question concerns the use of the quantifier "for some". My knowledge of quantifiers and formal logic is minimal, so I'm wondering if someone can explain to me intuitively why "for some" is used instead of "for all". For when I find the union of a set, aren't I including the elements from all the sets inside the set?

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  • $\begingroup$ You form the union $U$ of set $C$ collecting all the elements $x$ of the elements $X$ of the set $C$. Thus the "little" $x$ must be "inside" one of the "big" $X$ : this means "for some $X$ in $C$". If you say "for all $X$ in $C$", you will say that a "little" $x$ is selected for the union $U$ only if it belongs to all the elements $X$ of $C$. $\endgroup$ – Mauro ALLEGRANZA Aug 4 '14 at 18:05
  • $\begingroup$ This really helped. Thanks! $\endgroup$ – Nate Aug 4 '14 at 18:55
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Take the example of $A\cup B$ (or $C=\{A,B\}$ in this case). Then $x\in A\cup B$ if and only if $x\in A$ or $x\in B$ if and only if $\exists X\in\{A,B\}$ such that $x\in X$.

If you would have written "for all $X\in C$" you would get the intersection of all the sets in $C$, which can be everything if $C$ is empty, and that's not a set.

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    $\begingroup$ Asaf i think your excursion into non-well-founded set theory is an unintentional consequence of a shift key slippage $\endgroup$ – David Holden Aug 4 '14 at 19:07
  • $\begingroup$ Yeah, thanks. :) $\endgroup$ – Asaf Karagila Aug 4 '14 at 19:08

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