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So I gather that a set containing itself is not allowed. Yet it seems like a set of all concepts (Concepts) should contain an element denoting the idea of "concept". Is it that there is a difference of type re. "Concepts" and "concept"? Or is the element "concept" different in kind to other elements? Both seem unsatisfactory. Is this then a special case, and are there others?

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    In some Axiomatic Set Theories a "set containing itself is not allowed". There are others allowing it; see Non-well-founded set theory – Mauro ALLEGRANZA Oct 12 '20 at 11:21
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    There is a concept of a set, but the set itself is not a concept. So your set does not contain itself, not that there is anything wrong with that in principle. – Conifold Oct 12 '20 at 14:26
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    Notice the slip: At first, you treat concepts as extensional entities and make them members of a set. Then, treat them as intensional. There's nothing wrong about "concept of concept" just as there's nothing about "meaning of meaning." – Tankut Beygu Oct 12 '20 at 14:28
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    'Metaphor' is a meta metaphor. And meaning doesn't mean what we mean it to mean, because answers aren't the answer. – CriglCragl Oct 14 '20 at 13:01
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    Sure, but the concept of 3-apple set is no more a 3-apple set than the concept of apple is an apple. If you want self-reference so much why not simply say that the concept of concept falls under itself, without dragging sets into it. The concept of vague concept also falls under itself. And "word" is itself a word. But again, so what? – Conifold Oct 14 '20 at 14:33
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Mauro's first comment is a perfectly fine answer. I'll expand upon it.

In informal set theory, a set is an unordered collection of things. While a list has order and might have duplicates, a set either contains a thing or it does not. Informally, people could just define sets by describing which things are in them. Some examples:

  • A: The set of all integers
  • B: The set of one-digit decimal integers.
  • C: The set of subsets of B which contain three elements. For example, {2,3,4} and {0,6,8}.
  • D: The set of sets that do not contain other sets. B is an example.
  • E: The set of sets that do contain other sets. C is an example.

Unfortunately, this informal set theory led to complications, as Russell noted. The phrase "the set of all sets that do not contain themselves" appears to be a valid set description, but it is not, as that would lead to a paradox.

There have been multiple ways of formalizing set theory to avoid such problems. In some of them, sets cannot contain other sets, but in others, they can. See Mauro's comment on the question for links to more details.

A "set" is a "concept", so the "Set of concepts" would indeed contain itself, assuming it exists. If the formalization you are working with would not allow it, then the set does not exist. If it does exist, then it would certainly contain itself.

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