>>12607504>a set is something where you can ask a question "is x in the set" and get a yes or no answerThat is not a proper definition. You have described a property that a set has (you can check whether something is a member of it), but you haven't actually defined what a set is.

>inb4 muh zfcthose also describe properties that you want sets to have, it doesn't actually say what a set is.

ZFC leaves sets undefined:

https://mathoverflow.net/a/300814Description != Definition

If you want an example, all Euclid says about line segments is that given two points, you can join them to get a line segment. Is that a proper definition? I'm guessing you'd say no, since a property of line segments is that there have two endpoints; but just stating this does not define a line segment.

The proper way is probably by specifying coordinates for the two endpoints and computing the equation of a line between the two of them.

What Euclid did was to describe what a line segment is, but with the Cartesian system, we can actually define a line segment.