The history of set-theoretic definitions of $\mathbb N$ What representations of the natural numbers have been used, historically, and who invented them?  Are there any notable advantages or disadvantages?
I read about Frege's definition not long ago, which involves iterating over all other members of the universe; clearly not possible in set theories without a universal set.  The one that is commonly used today to construct natural numbers from the empty set is

*

*$0 = \{\}$

*$S(n)=n\cup\{n\}$
but I know that another early definition was

*

*$0=\{\}$

*$S(n)=\{n\}$
Unfortunately I don't know who first used or popularized these last two, nor whether there were other early contenders.

Maybe this book might be useful for you, too. I'll include a short quote from §1.2 Natural numbers.
Ebbinghaus et al.: Numbers, p.14

Counting with the help of number symbols marks the beginning of arithmetic.
  Computation counting. Until well into the nineteenth
  century, efforts were made to trace the idea of number back to its origins
  in the psychological process of counting. The psychological and philosophical
  terminology used for this purpose met with criticism, however,
  after FREGE's logic and CANTOR'S set theory had provided the logicomathematical
  foundations for a critical assessment of the number concept.
  DEDEKIND, who had been in correspondence with CANTOR since the early
  1870's, proposed in his book Was sind und was sollen die Zahlen? [9] (published
  in 1888, but for the most part written in the years 1872—1878) a
  "set-theoretical" definition of the natural numbers, which other proposed
  definitions by FREGE and CANTOR and finally PEANO'S axiomatization
  were to follow. That the numbers, axiomatized in this way, are uniquely
  defined, (up to isomorphism) follows from DEDEKIND'S recursion theorem.

Dedekind's book Was sind und was sollen die Zahlen? seems to be available online:
Wikipedia article on Richard Dedekind gives two links, one at ECHO
and one at GDZ. 
