On 29-Jul-2006, email@example.com (John Baez) wrote in message <firstname.lastname@example.org>:
> 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.
OK, but I'd be interested to know which ZF axioms your "imagine[d] reasonable people" don't believe. Or is their problem with mathematical logic?