In article <email@example.com>, Jim Heckman <firstname.lastname@example.org> wrote:
>On 26-Jul-2006, email@example.com (John Baez) >wrote in message <firstname.lastname@example.org>:
>> But as you might have suspected, not *all* ordinals can be written >> in this way. For one thing, every ordinal we've reached so far is >> *countable*: as a set you can put it in one-to-one correspondence >> with the integers. There are much bigger *uncountable* ordinals - >> at least if you believe you can well-order uncountable sets.
>? Is that last a reference to the Well-Ordering Theorem (equivalent >in ZFC to the Axiom of Choice)? Of course, you do need the WOT to >prove that /every/ set can be well-ordered, but ZF alone proves the >existence of uncountable ordinals.
That's interesting; I don't know if I ever knew that! The last time I really studied axiomatic set theory was decades ago.
Anyway, I can easily imagine reasonable people who are comfy up to omega or epsilon_0 (say) but don't believe you can well-order any uncountable sets. So, I didn't want to get into a fight by claiming bluntly that there *are* uncountable ordinals, without any sort of caveat. I didn't want to be advocating ZFC - but now that you bring it up, I don't even want to be advocating ZF.
But, I don't want to argue *against* them, either.
In fact, these days to get my back up you'd need to take a fairly drastic position, like my friend Henry Flynt, who argues that "mathematical knowledge amounts to the crystallization of officially endorsed delusions in an intellectual quicksand":