Re: I don't like the Axiom of Choice

Aatu Koskensilta wrote:

Indeed it is. My favourite counter-intuitive result that can be true in
absence of choice is that every infinite cardinal may be of cofinality
omega. Now there's a strange image!

Hmmm.

Surely this is only true for a limited view of what is a cardinal,
in the absence of choice? Surely there is more than one way
to define a cardinal, in that context. Your language above
seems to imply that every cardinal is still well-ordered,
but need this be so? Is there any "standard" convention
on this? (Seems unlikely, for such a non-standard matter.)

But surely we would still want to speak of "the cardinality of R"
in such circumstances? So shouldn't a cardinal number be closer
to Russell's definition - the class of all sets equinumerous to one.
Or, if you don't want to go near NGB classes, do the usual trick
and use regularity to pick out the set of all equinumerous sets
at the mimimal level.

-----------------------------------------------------------------------
Bill Taylor W.Taylor@xxxxxxxxxxxxxxxxxxxxx
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Set theory is a shotgun marriage - between well-ordering and power-
set.
The two parties get along OK; but they hardly seem made for each
other.
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.

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