[FOM] Infinite ordinals in Zermelo set theory

[Frode Bjørdal]

As all know, in contrast to von Neumann ordinals which are such that
a+1=au{a}, Zermelo has it that a+1={a}. It is nowadays common to say that
Zermelo
set theory (i.e. ZFC without replacement) cannot adequately deal with
infinite ordinals. For example, Zermelo set theory cannot prove the
existence of omega+omega if omega is taken as the first infinite von
Neumann ordinal.

But Zermelo did espouse a theory of ordinals, if I understand Akihiro
Kanamori, in http://www.math.ucla.edu/~asl/bsl/1501-toc.htm,
correctly. On p. 45 we read: "But in point of fact, Zermelo was most
probably the first chronologically to have formulated the concept of
ordinal, and this by 1915 in Z ˜ philosophy; urich. The rudiments of the
theory appear in items g--Tait in his Nachlass, 5 and indications are
there of collaboration with
Bernays. 6 ilosophy. With Zermelo never to publish this work, the
first published comments about it would appear in a later paper by
Bernays [4] pp. 6,10. 7 "

(I leave the references deconstructed here. Those interested may go to
Kanamori's paper.)

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.)


-- 
Frode Bjørdal, PhD (UC Santa Barbara)
Professor i filosofi
Institutt for filosofi, ide- og kunsthistorie og klassiske språk
Universitetet i Oslo
Kontor: Niels Treschows hus, rom 314
Telefon: 22844443