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In the famous paradox Russell tries to construct the set of all sets not containing itself. Then the question whether this whole set contains itself leads to a contradiction - if it does not contain itself then it should according to the definition, and if it does then it should not.

But let's consider the opposite construction - the set of all sets containing itself - and ask the same question: does this set contain itself?

It appears to me that both answers are consistent. If the set contains itself then it should (we are fine). But if it does not contain itself then it should not (we are also fine). We have no clue which option to prefer.

How should we interpret such a result?

  1. Does that mean that there are two such sets - one containing itself, another not?
  2. Or is it an example of a true statement which does not follow from any axioms (i. e. a manifestation of Godel's incompleteness) and we can choose either option as a separate axiom?
  3. Or is it still a kind of a paradox because a uniquely defined set has two instances?

Thanks in advance.

Ivan Rygaev
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    See the discussion at this MSE question and followup at MO. It is not in fact clear that either outcome is consistent (with some reasonable set theory allowing for such a thing in the first place)! – Noah Schweber Oct 28 '21 at 17:29
  • Thanks a lot!

    What about "This statement is true"? Again, naively it can be consistently both true or false and we have no reason to prefer either option. This seems to be a question of pure logic and does not require set theory. Does it depend on the definition of truth? I've seem a discussion on Lob's theorem and the question of "This statement is provable", but "This statement is true" seems to be a different matter.

     – Ivan Rygaev Oct 28 '21 at 18:39
  • So the reason we shift from "this statement is true" to "this statement is provable" is that per Tarski we can't express "this statement is true" in the necessary way in the first place (whereas Godel showed that we can express provability). So it's unclear how to pose that question at all. – Noah Schweber Oct 28 '21 at 18:47
  • Interesting. There is a parallel of Russell's paradox in linguistics which (unlike "This statement is true") sounds as a perfectly meaningful and reasonable question. You can call an adjective homological if it describes itself. So the adjective 'short' would be homological, because it is short. And the adjective 'long' would be heterological. Is 'heterological' homological - contradiction. Is 'homological' homological - both options possible. I suppose this problem also cannot be represented correctly according to Tarski since it also conflates syntax and semantics. – Ivan Rygaev Oct 28 '21 at 19:05
  • I previously asked a very similar question here. Perhaps some insight can be gained from the comments & answers of that question. – Graviton Oct 28 '21 at 23:17

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