Reference request for Zermelo's construction of natural numbers I have heard that back in 1908 Zermelo proposed to use $\emptyset,\{\emptyset\},\{\{\emptyset\}\},\ldots$ as the natural numbers. However, later von Neumann proposed the alternative approach of defining natural numbers in the following way,
$$0=\emptyset\\1=\{0\}\\2=\{1,0\}\\\vdots$$My question is,

Where can I find the detailed construction of natural numbers in the way Zermelo proposed? It will also be very good if there be a comparative analysis of the von Neumann construction and the Zermelo construction. 

I have searched internet for sometime but couldn't find anything relevant. Can anyone give me some reference regarding this? 
 A: The reason you are having difficulty finding such a reference is probably due to the fact that presentations of the natural numbers rarely deal with these construction directly. Instead we simply require an element $0 \in \mathbb{N}$ and a funciton $S : \mathbb{N} \to \mathbb{N}$ that gives the "successor" of each number. For the Zermelo naturals, $0 = \emptyset$ (as you mentioned) and $S(x) = \{ x \}$.
The typical construction proceeds by proving the Peano axioms for the given $\mathbb{N}$. The only part which will be significantly different in the case of Zermelo's naturals is proving that there exists a set $\mathbb{N}$ s.t. $\forall x \in \mathbb{N} \; S(x) \in \mathbb{N}$ (i.e. that there is a set which contains "all" the natural numbers). This requires invocation of the axiom of infinity. If you look at the definition of that axiom, you can see that the axiom in fact directly asserts that there exists a set which contains the von Neumann naturals, which is probably part of the reason presentations use this construction.
To prove the existence of such a set for Zermelo's naturals, the easiest way I can think of is establishing a bijection between the von Neumann naturals and Zermelo's, and then using the axiom schema of replacement.
The other interesting part of the Peano axioms is the induction axiom. The same proof that works for the von Neumann naturals (detailed here) will also work if we use the bijection between the Zermelo and von Neumann naturals again.
