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Is there a difference between properties and sets? To me, it would seem that the property of being non-self-identical is the same thing as the empty set, and the property of being (identical to x OR identical to y) is the same thing as the set {x,y}. So, what is the difference, if any, between properties and sets?

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    Intuitively, yes. Every property identifies a set: the set of all and only those objects satisfying the property. But...
    – Mauro ALLEGRANZA
    Commented Oct 25, 2020 at 17:07
  • 2
    ... Russell’s Paradox
    – Mauro ALLEGRANZA
    Commented Oct 25, 2020 at 17:07
  • +1 Excellent question!
    – J D
    Commented Oct 25, 2020 at 19:00
  • The property x = x does not characterize a set. Its extension is too big.
    – user4894
    Commented Oct 26, 2020 at 6:34
  • @user4894: That is not correct. "Too big" is no more than an artifact of some set theories. In alternative foundational systems that have a universal set/type U, the trivially true property does characterize U. The Russell property, on the other hand, does not characterize a set/type with boolean membership, but it may still characterize a set/type (depending on the system). It is worth noting is that the powerset/powertype in these systems are strictly smaller than U, because of Cantor's theorem. It is actually not size, but complexity.
    – user21820
    Commented Nov 14, 2020 at 14:51

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I suppose you are looking for reasons not to identify properties to sets.

(1) A set is a particular ( an abstract particular) , but properies are often considered as universals .

(2) A property is something an object possesses, shares; it is also the case for a set? I mean, could I say that an apple " possesses" the set of red objecs?

(3) Suppose you eat a red apple; the set of red things that existed before you ate the apple no longer exists; ti is replaced by a totally new set ( due to the extensionality principle) ; but the property " being red" remains unchanged.

(4) Two properties can be distinct in spite of the fact they are expressed by 2 terms referring to the same set. ( There is a conceptual / intensional aspect in the definition of a property).

(5) Not all properties determine a set , as is shown by Russell's paradox.

Reference : Marmodoro & Mayr, Metaphysics ( Oxford).

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