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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?

  • Look up Munchauseen trillema questions. This should put into context your question. The trillema observes three things: 1. All axioms are assumed. 2. All axioms are linear continuums. 3. All axioms circle to further axioms. Now relative to language as a starting point we can see this in a dictionary. 1. X word is assumed. 2. X word progresses to Y word with Y word being assumed. 3. Y progresses back to X, thus making a cycle while progressed to z. 4. Y and X as cycle progress to some other word. This is the nature of language. Now applying this to sets: A. 1 is assumed. B. 1 progresses to 2 th – Eodnhoj7 Aug 26 '19 at 19:43
  • I looked up Munchauseen trillema and how proofs may need subsequent proofs, but I still do not understand how that relates to my question. Any help? – Yukang Jiang Aug 26 '19 at 20:08
  • Yeah, replace word axiom with set. So one set. This set progresses to another set, with this new set looping to the original set as a new set which progresses...etc. You can find this basic nature in a dictionary. One word is assumed, this progresses to another word to define it, which progresses back to the original in a loop that is defined while progressing to a new word. The loop is defined by a new word, etc. – Eodnhoj7 Aug 26 '19 at 21:01
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    This question is more suitable for English SE. Natural language is often ambiguous, but there are no "problems" here, and there is nothing to "accept", really. These issues are settled by linguistic conventions. There are two different but related notions, both need to be labeled somehow, "proper part", and "strictly contained" are used when the convention is not clear from the context. Which labels to use for what is not a question about philosophy. – Conifold Aug 26 '19 at 22:14
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    @YukangJiang Your arm is not yourself. You shouldn't say "my arm is part of my arm" or "I am part of myself". – curiousdannii Aug 26 '19 at 23:47
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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.

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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?

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  • Yes, there are mathematical issues with a set being an element of itself, but given how much weight mathematical proofs have, I guess my question would be why doesn't a linguistic/historical point of view carry any weight at all (at least not that I've heard)? Ok maybe the treasure chest was a bad example, but if I asked a random person off the street if an empty jar contained anything, they would say "nothing, it's empty" or maybe "air;" they usually wouldn't say "an empty jar." How can an empty jar contain an empty jar? – Yukang Jiang Aug 26 '19 at 19:56

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