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In our lecture notes, the term "family" is used quite persistently and with no definition given. Some examples:

(i) Let V be a vectorspace and $(v_i)_{i \in I}$ a family of vectors... (Linear Algebra, Vector Spaces)

(ii) Let F be a family of balls $B = B_r(x) \subset \mathbb R^n$ ... (Measure Theory, Vitali Covering Lemma )

(iii) $(A_\iota)_{\iota \in I} \subset M, \ \exists g: I \rightarrow M$ s.t. for $\forall .. $ (Analysis, Axiom of Choice)

I was always assuming that this term is used to avoid talking about "sets of sets" with regards to Russells Paradox. Is this correct andor are there any further reasons?

Thanks

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Families are used in case (i) because sets do not suffice: one needs multisets (sets with multiplicity, i.e. repeated elements) when dealing with linear dependence, e.g. see this question. –  Math Gems Feb 27 '13 at 19:29
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up vote 11 down vote accepted

No, a family is not a term used to avoid proper classes. It is a term suggesting that all the objects (often sets) have something in common. A family can be thought as a function, e.g. a map from $i$ into a set $X$ (which itself can be a set of sets in some cases). But instead of actually caring about the function itself we care about its range and indexing. This is mathematically indistinguishable from caring about the function, but it allows the reader to put the emphasis on the objects rather than the function, its domain, and so on.

So we simply require that these objects have something in common, and they are a family. For example, they are all vectors from $V$, or open balls in $\Bbb R^n$, and so on and so forth.

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It is just another notion of function. That is a familiy $(v_i)_{i\in I}$ of vectors is a function $f\colon I\to V$, it's just that we write $v_i$ instead of $f(i)$. This differs from the set $\{v_i\mid i\in I\}$ as repetition is allowed and in fact the "order" is relevant, that is another family that differs only by interchanging some $v_i$ with $v_j$ is indeed a different family (if $v_i\ne v_j$). The term family of objects is used almost like set of objects, but compared to when we talk about the function the focus is more on the $v_i$ than on $I$ for example.

Compare with the notion of sequence, which is nothing but a function with domain $\mathbb N$, but puts a different focus.

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In my experience, the word "family" is many times used for sets of sets which are valid sets themselves, just to make it clearer that they are sets of sets.

The word I usually use when trying to sidestep Russel's paradox is "collection" ("The collection of all sets").

Wikipedia, however, says it's also used for collections of sets that are proper classes or multisets. (or multi-proper-classes, I guess).

It's probably a little different for every author / institution.

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