TITLE: What is the meaning of set-theoretic notation {}=0 and {{}}=1?
QUESTION [11 upvotes]: I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his head. So my question is pretty straight-forward, why 0={}, 1={{}}, 2={{{}}} and so on...
For example if we put it in a true statement about the natural numbers it doesn't make sense: 4×2=8 does it mean that {{{{{}}}}}×{{{}}}={{{{{{{{{}}}}}}}}}? The only concept of set "multiplication" is the Cartesian product but unfortunately I don't know it... Also if we have 6/2=3 does it mean that {{{{{{{}}}}}}}/{{{}}}={{{{}}}}? So there's some sort of set "division"?
Also why does one think that a "number" is a set?
Or is it that I'm the one who misunderstands things here maybe we "assign" a natural number to a set, that is for example: 1→{{}}, but then what does "assign" mean and how do you define the natural numbers in the first place?
Please clarify these concepts to me and I'll be deeply grateful to you :) Thanks!

REPLY [2 votes]: You learned when you were young that 0 is zero. But that's silly: 0 is a circle drawn on a computer screen, not a number!!
What you really learned is that 0 is a way to represent the number zero in a visual medium.
It is useful to have ways to represent numbers in a set-theoretic medium. The notation scheme typically used is to represent a natural number as the set of all smaller numbers; e.g. we represent five as $\{ 0, 1, 2, 3, 4 \}$. '3' here, of course, means our representation of three, which is $\{0,1,2\}$. The first few numbers in this representation if we really need to fully spell them out is

0 = {}
1 = {{}}
2 = {{}, {{}}}
3 = {{}, {{}}, {{}, {{}}}}
4 = {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}