3

When I’m referring to a set of sets, I like using the expression “a family of sets”, or “a collection of sets”, as is customary. But I was wondering, what if we needed to refer to a set of sets of sets? I have not found a specific word that identifies the third “layer”, as family and collection identify the second. Do you have any personal preference? I’d like to hear your suggestions.

Federico
  • 862
  • Personally I think introducing any new word is not worth the confusion. You could call it a collection of families of sets if you wanted to avoid repetition. – pancini Jul 08 '20 at 02:00
  • 1
    This definitely isn't universally true, but often if I have to refer to a collection of families of sets, the families have a name. So like "a collection of open covers." – DMcMor Jul 08 '20 at 02:04
  • "A set of families of sets" doesn't sound bad to me. – Jackozee Hakkiuz Jul 08 '20 at 02:05
  • Most set theories have no urelements, i.e; all (non-empty) sets are sets that contain sets that (may) contain sets... etc. Luckily we've adopted the term rank. The rank of a set is exactly what you describe as "layers". So a set of sets of sets can simply be called a set of rank 3 (or a rank-3 set). Related. – Graviton Jul 08 '20 at 02:08
  • @Graviton, that’s very useful information, thank you! I’m not really sure I’m satisfied with this terminology though. How would you translate “a set of collections of open subsets of $X$” in the language of ranks? – Federico Jul 08 '20 at 02:35
  • @Federico Unfortunately I don't think there's easy terminology for something so specific. Personally, I'd do something exactly like "Let $L$ be a set of collections of open subsets of $X$", then simply refer to $L$ from there on. I usually let the syntax or equations speak for me. – Graviton Jul 08 '20 at 02:41
  • 1
    @JackozeeHakkiuz Unfortunately the word "family" has a more specialised meaning than "set" -- it carried connotations of indexing, and the line then becomes blurred between a "set" and a "sequence". – Prime Mover Jul 08 '20 at 06:28

2 Answers2

1

There is no benefit to bringing in a whole bunch of synonyms for the same concept.

Children, when first learning to communicate, are force-fed a whole bunch of more-or-less arbitrary rules by people whose thinking is generally not very sophisticated (or they'd be doing something more intellectual than baby-sitting a bunch of small children), and these rules get embedded.

One of these rules is something like: "When you are communicating something which requires the same concept to be used more than once, never use the same word for that concept." While this may or may not be a good rule to use when writing some flowery novel or other ephemeral entertainment, it is in general not a good rule to use in technical writing.

The main problem with saying something like "a collection of families of sets" is that the terms "collection" and "family" (most particularly "family") have more specialised meanings than merely being a synonym for "set". A "set" is defined precisely by means of a precise axiomatic framework which, at base, is that it is defined solely by the elements it contains, and has specific rules (for example: the Zermelo-Fraenkel axioms) by which it may be constructed in order to specifically exclude constructions which lead to inconsisencies and paradoxes. A "family" calls to mind the idea that there is an "indexing set" which seeks to identify each of the elements with the elements of an auxiliary set which is defined independently. So unless you really mean "family", it is recommended that you don't use it. On the other hand, "collection" is looser than "set". It has (as far as I know) no rigorously formal mathematical definition, and just means "a bunch of stuff".

In conclusion:

"A set of sets of sets" is perfectly adequate, if this is really what you mean to write about. Mind, you should consider the question that if you need to go three levels deep, you may need to think about a recursive definition for whatever it is you are seeking to define.

Prime Mover
  • 5,005
0

I am bringing definition of family from Bourbaki Theory of sets: Function $f=(F,A,B)$ is defined by triple, where $A$, $B$ are sets, $F$ is functional graph and domain $pr_1F=A$. Functional graph can be called family, the domain is called index set and the range $pr_2F=B$ is called set of elements of family. Indicial notation $f_x$ is used to denote the value of $f$ at the element $x$.

zkutch
  • 13,410