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What is the proper name for "a set of sets"? Is it just a "higher-order set" in general or a "secondary set" in particular? A Wikipedia link would be great. I've been unable to find a special term for "set of sets" there.

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In modern set theory, every set is a set of sets. Anyway, what's wrong with "set of sets"? That's a fine term. –  Qiaochu Yuan Aug 8 '11 at 19:28
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Suppose that we make the usual set-theoretic definitions of natural number, then integer, then rational, then real. Then something as modest as $\sqrt{2}$ is a fantastically elaborate concoction of sets of sets of sets and so on. –  André Nicolas Aug 8 '11 at 20:02

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up vote 7 down vote accepted

As others have pointed out, technically, there is no distinction between a "set" and a "set of sets"; in fact, in most modern set-theoretic formalizations of mathematics, every object is a set, and so there is nothing a set can contain besides other sets. A set such as $\{\{\{\emptyset\}, \emptyset\}, \emptyset \}$ contains two sets, one of which contains two sets, one of which contains a set. In examples much more complicated than this, the notion of "second-order set" does not make much sense. And in case you don't believe such sets come up, see Arturo's comment below.

However, for psychological reasons, other terms such as "family" and "collection" are often used. (Royden's book Real Analysis describes a hierarchy for which terms he uses when; see the first paragraph of Chapter 1, Section 1 (p. 6 in my edition).) This is similar to the psychological reasons that we choose to use different symbols (e.g., $\cdot$ or $+$) for group operations, depending on what sort of group we're looking at.

One last technical point: Logicians do sometimes work with "languages" in which every object is, say, a real number rather than a set. In cases such as these, one needs second-order logic to talk about sets at all, and third-order logic to talk about sets of sets, etc. However, most mathematicians work, more or less, in the "first-order language of set theory": every object is already a set, so there is no need for higher order sets. (Although there are a few people who prefer second-order set theory as a formalization of mathematics, I think they are in a very small minority of the people who care about such things.)

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The set you give in your first paragraph is not a natural number, because it is not transitive: $\{\emptyset, \{\emptyset\}\}$ is an element, but not a subset. Either $\{\emptyset,\{\emptyset\}\}$ (which corresponds to 2) or $\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$ (corresponding to 3) would be natural numbers, though. –  Arturo Magidin Aug 8 '11 at 20:30
    
@Arturo: Corrected (with a reference to your comment, so please don't delete it). Thanks! –  Charles Staats Aug 8 '11 at 20:47
    
There is one more technicality that you forget to mention. A family might not exist as an object in the universe, but still can be defined externally. The "collection of all sets" etc. –  Asaf Karagila Aug 9 '11 at 15:13
    
@Asaf: Good point, although I think the word "collection" or sometimes "class" is used for that more often than "family." Anyone who wants to know about this should check out the Wikipedia article on "proper class"; I'm not going to try to explain it here. –  Charles Staats Aug 9 '11 at 22:56

In set theory, it is just a set. We do not distinguish between the levels, as the elements of a set are sets, but they could be sets of sets of sets of .... In fact, the usual ordinals have one set at each level (up to that ordinal)

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In some contexts, a "set of sets" is called a family. Some examples:

  • Extremal set theory: an intersecting family of sets.
  • Matroids: a family of independent sets (but: a set of bases).
  • Topology: a family of open sets.

This is semantic terminology. "Physically" (if you "believe" in ZFC) everything is considered a set, as mentioned by Ross and Qiaochu.

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I dislike the phrase "believe". You do not "believe" the axioms. You assume them, you work within their context, etc. Whether or not ZFC is consistent, trying to come up with nontrivial proofs (i.e. not deduced from contradiction) is a serious mathematical challenge. –  Asaf Karagila Aug 8 '11 at 20:08
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@Asaf: You have expressed a philosophical position that not all mathematicians hold. It's not a bad position, but it should not be stated as fact on a forum such as this one. –  Charles Staats Aug 8 '11 at 20:12
    
@Charles: I think that mathematicians that "believe" in their axioms are adding a religious dogma into mathematics. This should not be. Mathematics is not about belief but rather about logical inference. Either way, writing "believe in ZFC" is as degrading as calling all finitists morons, in my eyes. –  Asaf Karagila Aug 8 '11 at 20:19
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@Charles: I have a strongly negative view on anything which postulates any concrete and absolute answer. Be it in real life or in mathematics. That is not what is open for discussion here. I do not see the point to mention anything about ZFC when I sit down to work, and I am very much unworried by that issue. I still don't find any place for "belief" with regards to the consistency of ZFC or FOL or proposition calculus. Instead, I do my best to invest my time in mathematics, and the occasional pointless internet debate :-) –  Asaf Karagila Aug 8 '11 at 20:51
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What I meant was rather that (1) there are other foundations possible (e.g. topos theory) and (2) we need not care about foundations (e.g. we're using type theory). Everything is a set only if you want to think of matters this way. Even within the framework of set theory, there may be atoms. When I'm programming, I'm certainly not thinking of my integer variables as Cantor (or Church) ordinals (plus sign). –  Yuval Filmus Aug 8 '11 at 21:09

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