If a set is a group of objects, then what is an object?

Question

If a set is a group of objects, then what is an object?

My best try at this is the following:

An object is anything that we can discuss or think about, separately from everything else. It is not necessarily a physical object, it can also be an imaginary object such as a green elephant.

Once we have defined an object, then it makes sense to talk about groups of objects, and this is where set theory begins.

Answer 1

From Joseph Shoenfield, Mathematical Logic (1967), page 238 :

a set $A$ is formed by gathering together certain objects to form a single object, which is the set $A$. Thus before the set $A$ is formed, we must have available all of the objects which are to be members of $A$. [...]

We are thus led to the following description of the construction of sets. We start with certain objects which are not sets and do not involve sets in their construction. We call these objects urelements. We then form sets in successive stages. At each stage we have available the urelements and the sets formed at earlier stages; and we form into sets all collections of these objects.

A collection is to be a set only if it is formed at some stage in this construction [emphasis added].

We can carry out this construction with any collection of urelements. If we carry it out with no urelements, the sets which we obtain are called pure sets. It turns out that these are sufficient for mathematical purposes; and they are also sufficient to illustrate all the problems which arise in the general case.

[Thus, in the "standard" treatment of set theory, like $\mathsf {ZF}$] we shall therefore restrict ourselves to this case, and henceforth take set or class to mean pure set.

Thus, if our choice is to have urelements, we can start with a collection of objects whatever: physical or abstract ones.

But if we start with physical objects, we are not licensed to assume the existence of infinitely many of them, while some sort of "axiom of infinity" is necessary for the development of "current" mathematics.