From Set-theoretic definition of natural numbers
n+1=n $\cup$ {n}
i.e.
- 0 = {}
- 1 = {{}}
- 2 = {{},{{}}}
- etc
It seems to me that a simpler, equally valid definition would be
n+1={n}
i.e.
- 0 = {}
- 1 = {0} = {{}}
- 2 = {1} = {{{}}}
- etc
Why is the ZF definiton preferred over mine?
(The only "advantage" I can think of is that in ZF, $|n|=n$, but this strikes me as being circular reasoning.)