How to prove by induction that the set of all natural numbers is an ordinal I have seen alternative methods of this proof, with one being: let $n$ be the set of all natural numbers. Then
(1) $\omega$ is an ordinal, (2) If $\alpha$ is an ordinal and $\beta \in \alpha$, then $\beta$ is an ordinal. Therefore since $n \in \omega$, it follows that $n$ is an ordinal. (I'm not sure if this is definitely correct but this is what I worked out)
I wanted to know how to prove this by induction instead. I have so far, using standard induction format, that I need to prove (i) the empty set is an ordinal, and (ii) if $n$ is an ordinal then $n$ $\cup$ $\{n\}$ is an ordinal. 
I am not sure how to explicitly prove (i) and (ii) so any help would be greatly appreciated. Thank you.  
 A: There are several equivalent formulations of what it means to be an ordinal. For example:


*

*$x$ is an ordinal if $x$ is a transitive set which is well-ordered by $\in$.

*$x$ is an ordinal if it is a transitive set that all its elements are transitive sets.

*$x$ is an ordinal if all its elements are ordinals, and $x$ is transitive (note that there is an induction hiding in this definition, and it works best if you already have some definition of "ordinal" first).


So now to prove that $\omega$ is an ordinal you need to pick the definition that you want to use, and show that $\omega$ satisfies this definition. How to do so depends on the definition itself.
If you pick the third definition, for example, then you can prove by induction that indeed every $n$ is an ordinal, then prove that $\omega$ is a transitive set.
If you pick the first definition, then you need to prove that every two integers are comparable by $\in$ and that $\omega$ is transitive. This can be done by induction.
If you pick the second definition, then you need to prove that every finite $n$ is a transitive, then show that $\omega$ is transitive. Again, you can do this by induction.
So the point is that in order to prove that $\varnothing$ is an ordinal, etc. you need to pick a definition of an ordinal, and then show that it holds for $\varnothing$ (all of them hold pretty much vacuously), and then by induction prove that if $n$ is an ordinal, so is $n\cup\{n\}$. And this depends on the definition of an ordinal.
