Descendent of a Set: Find Definition & Symbol

[Rasalhague]

Given a set, A, is there a word and/or symbol that means "an element of A, or an element of an element of A, or an element of an element of an element of A, or..."

By analogy with the use of the term in linguistics, I wondered if descendent might be used in this way. The only hit I found on google for the expression in the title of this post looks a bit too technical for me at the moment.

http://projecteuclid.org/DPubS?verb...e=UI&handle=euclid.nmj/1118801525&page=record

I can't tell what its definition means, but I suspect it's using the word descendent in a different sense. Incidentally, does $\in \in$ mean "is an element of an element of"?

 

[CRGreathouse]

A good symbol (by analogy with reduction sequences) might be $$a\in^*A$$, the transitive closure of $$\in$$.

 

[jgm340]

What's your reason for wanting this symbol?

While you can certainly have sets of sets of sets of... etc., there is almost always an important distinction between an element of a set and a set (whether or not that element is a set itself). It seems as though you are looking for a way to compare very different things.

Anyway, I would probably do something like this:

Suppose A is your set.

Let A₀ := A, and A₁ be union of all sets in A₀. Likewise, let A_j be the union of all sets in A_j-1.

Let B be the union of all A_j for j = 0,1,2,...

Then to ask what you want above, simply ask: is x in B?
​

Now, I'm not sure if it is legal to define a set B like that. It feels really strange to have B be a countable union of sets...

 

[Rasalhague]

Thanks for the suggestions. The reason I thought of this was that a particular mathematical structure, such as a group, field or vector space, is defined as a tuple, and a tuple is defined using sets: (a,b) = {{a},{a,b}} or, if the possibility of a 0-tuple is admitted, (a,b) = {{{null},{null,a}},{{{null},{null,a}},{b}}}. In this example, a and b, the terms/entries/items of the tuple/sequence/list, are often called its elements or members, even though they aren't technically its elements if the tuple is viewed like this as a set. And people usually talk about, say, vectors as being in a vector space, or being elements of a vector space, even though they aren't elements of the tuple viewed as a set, or even terms of the tuple, but rather elements of one of its terms, namely the "underlying set" of the structure.

I thought it would be handy if there was a word that would blur the distinction, that could be used by someone who wants to be rigorous but not finicky to the point of distraction, a word that would capture the intuitive, transitive idea of containing and could be used if you don't want to specify exactly what place something has in the hierarchy of sets and subsets.

 

[Landau]

Now, I'm not sure if it is legal to define a set B like that. It feels really strange to have B be a countable union of sets...

Sure it is legal; nothing wrong with a countable unions of sets (it is itself countable). But there is another problem with the countability here: you don't know that our set A has only countably many "layers" of sets, so the definition is not general enough. But I'm sure it can be done with some sort of transfinite induction.