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There is a version of set theory according to which there are two flavors (types? categories?) of elementhood relation, and if it's ultimately coherent, it does offer a solution to Russell's paradox (with perhaps paradoxical consequences of its own).

But so imagine that being-an-intensional-element-of is one such relation, with being-an-extensional-element-of is another. (More finely, something is extensionally an element if it is intensionally an element, but not vice versa.) So we would not ask just, "Is there a set of all noncircular sets?" and, "Is that set not an element of itself?" but, "Is that set an intensional or extensional element of itself?" So that it could be self-intensional but not self-extensional, or self-extensional but then only indirectly self-intensional (perhaps this idea means a broader ramification of the auxiliary elementhood "types").

So to say (for i and e = intensional, extensional):

  1. ∀ab∃X
  2. If any a ∉i any b
  3. Then a ∈e X
  4. So if X ∉i X, then X ∈e X?

Alternatively (maybe more reasonable):

  1. ∀ab∃X
  2. If any a ∉e any b
  3. Then a ∈i X
  4. So if X ∉e X, then X ∈i X?
Kristian Berry
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  • You are misusing the word "intensional" in the question. In the linked section, the two membership operators are both extensional. – David Gudeman Jan 19 '23 at 13:51
  • @DavidGudeman I don't mean to directly represent DEST in this way, but to either (a) compare DEST to the question of a pair of intensionally different elementhood relations or (b) reinterpret intensionality itself as a diversification of the concept of extensionality. The theme is to find a "justifier" of DEST that isn't a seemingly *ad hoc* solution to Russell's paradox, I guess. – Kristian Berry Jan 19 '23 at 15:16
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    OK, but I don't think this use makes sense. Let E and e represent extensional and intensional membership resp. Suppose for all x, x E S1 iff x E S2. Then S1=S2, so there can't be any y such that y e S1 and not y e S2. So how can E be different from e? – David Gudeman Jan 19 '23 at 20:09
  • @DavidGudeman, [Hyperintensionality](https://plato.stanford.edu/entries/hyperintensionality/) might be relevant: "*H* is hyperintensional insofar as *H A* and *H B* can differ in truth value in spite of *A* and *B*’s being necessarily equivalent." Alternatively, if the = sign itself can vary with the different axiom of extensionality in play, perhaps we would not say that S1 and S2 were (completely) equivalent? – Kristian Berry Jan 19 '23 at 21:47
  • Majkić has "intensional elements" in [Conservative Intensional Extension of Tarski's Semantics](https://www.hindawi.com/journals/aai/2013/920157/), but I do not think they are what you have in mind. They are element-valued functions on possible worlds that specialize to ordinary ("extensional") elements in every possible world. – Conifold Jan 21 '23 at 11:03

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