I recently landed on a book written for elementary school teachers which introduced the concept of whole numbers in the following manner:
We have a set $\{\alpha, \beta, \gamma\}$. There are other sets we can identify that share a common property with this set by virtue of which a one-to-one correspondence can be established between the elements of the set $\{\alpha, \beta, \gamma\}$ and all other sets identified. This common property abstracted away from all these sets including $\{\alpha, \beta, \gamma\}$ and considered as a thing in and of itself is named '3'.
I've also come across some books on set theory that define 3 as literally the set {0,1,2} or {{}, {{}}, {{}, {{}}}}.
So now I'm confused what is the correct way of thinking about whole numbers? A property abstracted from sets and considered to be a thing in itself or literally as sets themselves (as in 3={0,1,2}?
The first approach appeals to me better because it is in line with what I think of 3. However, the second one, says 3 is a set, that's not what my intuition tells me. 3 and the set {0,1,2} are clearly different things.