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?

• They are not identical: $\{\{\phi\}\} \not=\{\phi,\{\phi\}\}$ – Tom Oldfield Oct 31 '12 at 14:40
• How many elements do the sets that represent $2$ have in each representation? – Mariano Suárez-Álvarez Oct 31 '12 at 14:40
• The second representation lets you also represent infinite ordinals. It is unclear what the "limit" of $\{\},\{\{\}\},\dots$ would be. You can quickly calculate the maximum and minimum of a pair of natural numbers in the second representation, and it is easy to define $\leq$ on those natural numbers using set inclusion. – Thomas Andrews Oct 31 '12 at 14:44
• Amplifying on Thomas Andrews' comment, the "limit" of Zermelo's sequence would have to be a set that is nested infinitely deep. Although some versions of set theory do allow such infinitely deep nesting, ZFC does not; the axiom of regularity is specifically designed to forbid this. – Ben Crowell Oct 31 '12 at 15:55