[FOM] Formalization Thesis

[Bill Taylor]

William Tait <williamtait at mac.com> asked:

-> One challenge to the Formalization Thesis, or maybe better, to a  
-> satisfactory formulation of it, arises from the 'reduction to sets' of  
-> various types of objects: ordered pairs, functions, ordinal numbers,

I have wondered for some time:-

IS THERE any formal treatment of ordinal numbers, or even just
countable ordinals, that does NOT rely on some set theory underneath it?

It seems hard to see how it could, given the limit-ordinal concept.

Is there a nice treatment anywhere?

Bill Taylor