Linguistic explanation of a set being an element of itself or containing itself and can you be part of yourself or contain yourself?

Question

Set theories examines whether a set can be an element of itself or contain itself.

But linguistics already offers its own explanation for whether a set can be an element of itself or contain itself.

If I asked you do you want the whole part of the apple or a small part of the apple, notice that a "part" can be the whole part. This leads to the conclusion that an apple can be a part of itself, because the whole part exists. Edit: The part of the apple is not distinct from the apple. I can say I bit a small part of the apple or the part of the apple is a small part of itself. So I would say from a cultural perspective (English-speaking cultures), that a set should be able to be an element of itself. My arm is part of myself. Therefore, the whole part is also part of myself.

Similarly, if there was a treasure chest, and inside the treasure chest was treasure and I asked you if the treasure chest contains treasure, you would probably say "yes." But, if I asked does the treasure chest contain a treasure chest, you might probably say "no." Going by this example, a treasure chest doesn't contain a treasure chest if only treasure (and air) were inside of it. A treasure chest wouldn't contain itself. How could an empty jar contain an empty jar?

So from a linguistic/cultural perspective, the answers to the problems above should be a set can be an element of itself, but it can't contain itself. And I would say I, as an individual, am part of myself, but I do not contain myself.

Why can't we simply accept the linguistic/cultural perspective over the issues stated above?

Answer 1

The first thing to note is that parthood is a better proxy for subsethood than for elementhood. E.g. as you observe, any object is part of itself (in the weak sense); this reflects the fact that any set is a subset of itself. Elementhood is a more finicky thing, though. In particular, subsethood is transitive while elementhood isn't, and that matters when we consider different "types" (e.g. the set S of all topologies on a set X is a set of sets of subsets of X, so elements of X are elements of elements of elements of S, but they're certainly not elements of S - a topology is more than just a single point!).

Now you might still ask why we don't form a separate theory based on "parthood" anyways. Maybe it doesn't do the same thing as set theory, but so what? The answer is: we do (or at least some of us do)! There is a subject called mereology which is exactly this. For a comparison of mereology and set theory, see here. However, a key point here is that mereologies often lack a lot of the structure that makes set theory useful.

That said, see this mathoverflow question for a discussion of the possible role of mereology in foundations of mathematics.

And of course none of this addresses the broader question, "Can we get motivation for/against self-containing sets from outside mathematics, or at least outside set theory proper?" That's an interesting question, but far too broad. Briefly, though, I'd say that the answer is yes, and indeed there's good reason to consider self-containing sets as totally valid kinds of objects, with ZF-style set theory amounting to the study of the "well-founded core" of the set-theoretic universe. But that's going far afield.

Answer 2

The main issues pertaining to what a set may contain or what a set may be contained in are mathematical, not linguistic. Put simply, from a standard understanding of a set as a mathematical 'container' of mathematical 'objects', allowing a set to contain itself generally produces problems with the resulting mathematics. That said, there IS a non-standard approach to set theory in which sets ARE allowed to contain themselves (I am not familiar with the details, but I know that there sets are given a graph-theoretic interpretation, where a set containing itself would be interpreted as a directed-edge from a node to itself).

On another note: Paraphrasing a line from a work of fiction I have read, why shouldn't a treasure chest contain itself? If not, what DOES contain it?

Answer 3

Simply an example, somewhat twisted, "The set of intellectual abstractions created by human beings" is both an element of itself, and part of itself, in a sense.