[FOM] Infinite ordinals in Zermelo set theory

Robert Black mongre at gmx.de
Sun Feb 8 04:24:37 EST 2009

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.


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