# How is the normal ordering on the Natural Numbers defined in Zermelo set theory?

A fact that often gets mentioned in the elementary development of arithmetic in ZFC is that there are a bunch of different ways one could have defined the natural numbers. The most common alternative to the modern way of doing things is due to Zermelo, who sets $0=\emptyset$ and $n'=\{n\}$.

My question is how does one define the normal ordering on this collection using only the resources Zermelo allowed himself (i.e. The normal ZFC axioms, minus replacement, foundation, and a modified version of the axiom of infinity which states the existence of the above set rather than the first von Neumann ordinal)?

• I'd guess by induction: $n\leq n$ and if $n\leq m$, then $n\leq m'$. – Wojowu Apr 10 '16 at 14:39
• Or one would try to build the transitive hull of $x<\{x\}$ – Hagen von Eitzen Apr 10 '16 at 14:40
• Yeah, but how do you actually carry it out? Not sure how to do this without the normal ordinals or replacement... – I. Boon Apr 10 '16 at 14:49

$$a < b =_{\text{def} } \forall w \ [\forall x \ (a \in x \to x \in w) \land \forall x \forall y \ (x \in w \land x \in y \to y \in w) \to b \in w].$$