# Are sets in ZFC a primitive notion?

I read everywhere (including here on math.stackexchange) that the notion of set in ZFC is primitive. To my (probably mis-)understanding, though, a primitive notion is a concept that is not defined in terms of previously-defined concepts.

This is where my confusion begins. Aren't ZFC axioms there exactly to define what a set is (in first-order logic terms)? Correct me if I am wrong, but isn't something a set exactly if and only if satisfies those axioms? With those at hand, one can for example show that {1,2} is a set but the collection of all sets is not a set.

To my very uneducated brain, this is what you do with any other definition in math: you say that something is something if and only if it satisfies certain axioms.

Therefore I am wondering: how does a set differ from say, a measure of probability? For both of them, I have a bunch of axioms that formally define what they are.

I do see that $$\in$$ and $$=$$ are primitive notions because they are indeed never defined but simply appear in the ZFC axioms. However, sets do. What am I grossly misunderstanding?

• The ZFC axioms specify how sets behave. They do not tell us what sets are. Jun 20, 2022 at 16:20
• But what's preventing me to say that something is a set if and only it behaves like a set? Jun 20, 2022 at 16:29
• If a set with a binary relation satisfies ZFC, then it's a model of ZFC, and its elements are the 'sets' in that model. Jun 20, 2022 at 16:59
• ZFC does not include a formal definition of the form: $X$ is a set iff ______________. It seems such a definition is simply not required. Jun 21, 2022 at 2:04