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Collection vs. set of subsets vs. set of all subsets?

These get mixed. But the idea that I have is:

collection: a set of subsets, where order is ignored. So it's like a weaker form of set.

set of subsets: could be a collection, however since this is set of subsets, then one may not know whether the weakened properties of collections suffice. Or whether one actually has set of collections.

set of all subsets or the power set: this could be a collection, but it's not required to be one, because one could refrain from using the term collection and rather use subset.

family of sets or family of subsets is by definition a collection $F$ of subsets of a given set $S$. So it seems like here collection is the same as set of subsets, but also family of subsets. Or also, family is a set.

I wonder what sense does it make to use the terms collection and family at all? If they are sets, then why not call them sets?

mavavilj
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  • Not every collection is necessarily a set. The collection of all sets, for example, is not a set in ZFC. – Daniel Mroz Sep 14 '18 at 19:20
  • It is only a terminology issue: in set theory we have only sets. See Halmos, Naive set theory, page 34. Thus, a family is a function, i.e. a set. – Mauro ALLEGRANZA Sep 14 '18 at 19:23
  • @DanielMroz So indeed a collection is a weak form of a set? That satisfies some constraints, but not all required for a set. However, is there a common definition for collection? Maybe it can also mean that "one wants to take a set of these, but cannot or doesn't need to assume that the constructed set has all set properties, thus it's a collection (which does not need to be a set)". – mavavilj Sep 14 '18 at 19:27
  • You may find this post insightful. – Daniel Mroz Sep 14 '18 at 20:01

2 Answers2

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In Axiomatic Set Theory everything is a set; it's the only kind of object the language talks about. So there you have sets of sets, sets of sets of sets, etc. For my own sanity I use family, collection, congeries as synonyms for `set'. So, I have some set where it all happens, it has subsets, there are families of (sub)sets and, if need be, congeries of families of (sub)sets. But, to repeat, these are all synonyms; the words serve to remind me at what level I am.

hartkp
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I think that you are radically over-analysing the language. A "collection" of (sub)sets, a "family" of (sub)sets, and a "set" of (sub)sets are all referring to the same kind of object: a set whose elements are subsets of some other set. Note, even, that the Wikipedia disambiguation page for "collection" links to "set" and "family of sets". Wikipedia does seem to make some distinction between a "family of sets" and the other terms, but this is likely highly context-dependent, and such a distinction probably wouldn't matter to someone who is not an expert in set theory.

As examples:

  • The powerset of a set $S$ is set (or collection of sets, or family of sets). In particular, the powerset $\mathscr{P}(S)$ is a set where every element of $\mathscr{P}(S)$ is a subset of $S$. Indeed, we can say something stronger: $$ A \in \mathscr{P}(S) \iff A \subseteq S. $$ In other words, the powerset of a set $S$ is a set (or a collection, or a family) which contains all of the subsets of $S$.
  • A topology on a set $T$ is a set $\mathscr{T}$ where every element of $\mathscr{T}$ is a subset of $T$. That is, if $U \in \mathscr{T}$, then $U \subseteq T$. For example, if we let $\mathscr{T}$ denote the usual topology on $\mathbb{R}$, then the interval $(0,1)$ is a subset of $\mathbb{R}$ (as every element of $(0,1)$ is a real number), but $(0,1)$ is a singleton element of $\mathscr{T}$ (since $(0,1)$ is an open set in the usual topology).
  • A $\sigma$-algebra on set $X$ is a set $\Sigma$ where every element of $\Sigma$ is a subset of $X$.

I suspect that the terms "family" and "collection" exist as a form of disambiguation. "A set of sets" is somewhat awkward to say, and in verbal communication I can see where it would be useful to distinguish between the underlying set, and the set which contains subsets of the underlying set (e.g. we can refer to a topology as a "collection", then talk about element of the collection and subsets of the underlying set).

Xander Henderson
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  • "In mathematics, the term "collection" is generally used to mean a multiset, i.e., a set in which order is ignored but multiplicity is significant." http://mathworld.wolfram.com/Collection.html – mavavilj Sep 14 '18 at 19:30
  • Or perhaps they are indeed all sets and one would use "ordered set" etc. tor higher forms of sets? – mavavilj Sep 14 '18 at 19:32
  • I don't claim to know everything, and I am not an expert on axiomatic set theory, but I have never seen that usage before, and have encountered the term "collection" to refer to a "set of sets" (again, the examples that I provided are concrete examples where I have seen a set of set referred to as a collection or family). Again, I am an analyst, and so something of a non-expert, but I think that the vast majority of mathematicians are similarly non-expert, and wouldn't make a distinction. – Xander Henderson Sep 14 '18 at 19:34
  • But as Daniel Mroz wrote in the comments, a collection may fail to be a set in ZF(C) depending on whether it doesn't satisfy some of the axioms. Clearly a collection, given the Wolfram def., would not satisfy Well-ordering. – mavavilj Sep 14 '18 at 19:34
  • @mavavilj Daniel Mroz could just as easily have said that "the set of all sets isn't a set." The sentence he wrote would have made just as much sense, and meant exactly the same thing. These terms are all completely synonymous (with the exception, perhaps, of a part of mathematics with which I have little to no contact). I can't speak for everyone, but I would imagine that most work-a-day mathematicians would agree with what I have written. – Xander Henderson Sep 14 '18 at 19:37