The thing about ZFC is that it doesn’t feel like “the definition” of a set. It seems like the notion of a “set” or a “property” came first, and then we came up with ZFC as a way of approximating that notion. There are statements about sets that are independent of ZFC, and this seems more like a shortcoming of ZFC than a shortcoming of the very concept of a set; perhaps we could come up with a philosophical definition of the word “set” that pins the concept down precisely, even if it means resorting to subjective concepts like simplicity or usefulness.
On the other hand, the word “set” doesn’t seem to be as well-defined as we would like it to be. I doubt that there is one unique concept that you could call “the set of all real numbers”, since this concept behaves different ways in different set theories, and I see no basis on which to say one set theory or another is “incorrect”.
The thing about ZFC is that it doesn’t feel like “the definition” of a set. It seems like the notion of a “set” or a “property” came first, and then we came up with ZFC as a way of approximating that notion. There are statements about sets that are independent of ZFC, and this seems more like a shortcoming of ZFC than a shortcoming of the very concept of a set; perhaps we could come up with a philosophical definition of the word “set” that pins the concept down precisely, even if it means resorting to subjective concepts like simplicity or usefulness.
On the other hand, the word “set” doesn’t seem to be as well-defined as we would like it to be. I doubt that there is one unique concept that you could call “the set of all real numbers”, since this concept behaves different ways in different set theories, and I see no basis on which to say one set theory or another is “incorrect”.