Q:

Better representaion of natural numbers as sets?

Natural numbers can be represented as
$0=\emptyset$
$1=\{\emptyset\}$
$2=\{\{\emptyset\}\}$
$...$
or as
$0=\emptyset$
$1=\{0\}=0\cup\{0\}$
$2=\{0,1\}=1\cup\{1\}$
$...$
What are the names of these representations?
Aren't they identical?
What are advantages of second representation?

A:

These are respectively Zermelo's and von Neumann's representations/implementations of the naturals in set theory.
If you just want to reconstruct arithmetic and then classical analysis inside set theory, Zermelo's representation works just fine and some standard textbooks do things that way.
But once we go beyond, and want to deal with infinite ordinals as well as finite ones, then you need a von Neumann-style representation. So many authors use it from the start, even for finite numbers.

