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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?

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    $\begingroup$ The ZFC axioms specify how sets behave. They do not tell us what sets are. $\endgroup$ Jun 20, 2022 at 16:20
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    $\begingroup$ But what's preventing me to say that something is a set if and only it behaves like a set? $\endgroup$
    – user332582
    Jun 20, 2022 at 16:29
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    $\begingroup$ 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. $\endgroup$
    – Berci
    Jun 20, 2022 at 16:59
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    $\begingroup$ ZFC does not include a formal definition of the form: $X$ is a set iff ______________. It seems such a definition is simply not required. $\endgroup$ Jun 21, 2022 at 2:04

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I think the narrative that sets are a primitive notion in ZFC is slightly incorrect. A more precise statement is that ZFC never discusses a notion of “set” at all. Instead, what is discussed is “set membership”; that is the primitive notion. Saying that ZFC defines “what a set is” or “how a set behaves” is, in my view, not correct.

At best, ZFC describes how the collection of all sets behaves.

An analogy would be the axioms of a group. These axioms define how the group itself behaves. They do not talk about the nature of group elements; they only discuss how group elements are related to each other by the group operation.

Similarly, ZFC describes how a universe of sets should behave. It does not discuss the nature of individual sets; it only discusses how sets are related to each other by the set membership predicate.

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  • $\begingroup$ Thank you for attending me on this, Mark. Could you tell me what a "primitive notion" is (or ironically how it is defined:-)) in your view. After all the question (is set a primitive notion?) can only be discussed if this is clear for every participant of the discussion. For me it is a concept that is undefined. You could compare it with an element that has no predecessors wrt a well-founded relation. In what sense does this differ from your conception of primitive notion? $\endgroup$
    – drhab
    Jul 23, 2022 at 7:01

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