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The usual addition function on integers has "signature" or "type" $+: \mathbb Z \times \mathbb Z \to \mathbb Z$. Similarly, one could try and write the "signature" of set union as $\cup: \mathbf{Set} \times \mathbf{Set} \to \mathbf{Set}$ where $\mathbf{Set}$ is the set of all sets ... except that's nonsense because it's not a set and therefore, I presume, the union (and intersection, cartesian product etc.) are not functions?

I can see that for any fixed universe-set $U$ one can define union etc. as functions on the powerset of $U$. But what kind of thing is "The Union" in general? Does this question even make sense? It seems to be one of the first concepts you encounter in elementrary set theory, and one of the few as far as I can tell that you can't actually formally define as a set.

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    $\begingroup$ Union and intersection, cartesian product are class function. More precisely, union is a (class) function that has a domain $V\times V$ (where $V$ is class of all sets. Note that $V$ and $V\times V$ are proper class, that is, these are not a set.) $\endgroup$ – Hanul Jeon Mar 5 '14 at 11:20
  • $\begingroup$ If you regard a function as set, $\cup$ is also proper class. (In other words, $\{(x,y,z):x\cup y=z\}$ is a proper class.) $\endgroup$ – Hanul Jeon Mar 5 '14 at 11:26
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Due to the paradoxes of Cantor's original set theory (Cantor himself found one, then Burali-Forti and Russell : this one is the more famous), mathematicians has been forced to introduce "restrictions" - through axioms - to the "natural" ways of conceiving sets.

Those restrictions have produced a certain "gap" between what is our more or less intuitive conception of set and the mathematical rigorous concept of it.

Our intuitive concept of set is a mixture of two "natural" ideas :

(i) the concept of a set as a collection of things : cristal clear when we think at a finite collection of things: we simply enumerate them (think at the $\{ a, b, ... z \}$ symbols).

(ii) the concept of a set as the extension of a property or predicate, i.e.the set of all red things (think at the set $\{ x : P(x) \}$ ).

Axioms of set theory "dictate" the rules about what are you allowed to call "set" and about which are the allowed rules you can use in order to build sets.

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