# 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?

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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-Alvarez 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