Re: is there a set of all ordinals

On Feb 28, 5:44 pm, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
you wrote
"n Z set theories, there is no set S such that every ordinal is a
member of S.

In ordinary class theories, there is no set S such that every ordinal
is a member of S, but there is a proper class C such that every
ordinal is a member of C"

i am not talking about what set theories or ordinary class theories allow
i am talking about
is there a set of all ordinals

I have no definitive opinion on the existence of a set of ordinals
outside of the context of a given theory or some class of theories.

obviously such a set was accepted in Burali-fortis day as they came up
with the paradox
did canto accept such a set
which since set theory has  now been denied only by introducing an axiom  

On the historical question of Cantor himself, I don't recall exactly
from my reading. Cantor did take certain collections (or whatever
German word he used), such as the set of all sets and its cardinality
(if I recall correctly) to be "inconsistent entities" (or some
expression like that), but I don't know whether a set of all ordinals
was one of them. However, I do know that Cantor was aware of the
Burali-Forti paradox by 1899 (Burali-Forti discovered the paradox in
1897, I think), so that's some reason to think that Cantor by that
time did regard a set of all ordinals as an inconistent entity.

Your best bet for books on Cantor are Hallett's book and Dauben's
book, the former being more about the philosophy of Cantor's
mathematics and the latter tending more toward Cantor's biography.
Also, Lavine's book is good on comparing Cantor with Zermelo and with
Fraenkel.

By the way, there's a new biography of Zermelo just published.

MoeBlee
.

Relevant Pages

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