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Natural numbers can be represented as





or as





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
up vote 5 down vote accepted

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.

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So first is Zermelo's and second is Neumann's? – Suzan Cioc Oct 31 '12 at 15:02
@Suzan Yes indeed – Peter Smith Oct 31 '12 at 15:40

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