[FOM] Infinite ordinals in Zermelo set theory

[Robert Black]

Frode Bjørdal wrote:
> Apart from an historic interest, my main, and systematic, interest in this
> is whether (and if so how, and how to do it most elegantly/optimally)
> Zermelo set theory will be strong enough to account for infinite ordinals
> in some other sense than von Neumann's, e.g in some sense related to
> ordinal notation. Such a question seems relevant as e.g. Saunders Mac Lane
> has  been on record stating that bounded Zermelo may suffice as a
> foundation for mathematics. (If I remember correctly, Adrian Mathias has
> stated that Mac Lane was not fond of von Neumann ordinals.)
>
>   
If you don't have replacement and want plenty of ordinals, define them 
using Dana Scott's trick as equivalence classes of orderings *of lowest 
possible rank*. This is done in Michael Potter's book _Set Theory and 
its Philosophy_. It obviously gives you all the ordinals below beth_omega.

Robert