
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted:
Jul 31, 2006 5:46 PM


On 29Jul2006, baez@math.removethis.ucr.andthis.edu (John Baez) wrote in message <eag2eh$bcq$1@news.ks.uiuc.edu>:
> In article <ead71n$eth$1@news.ks.uiuc.edu>, > Jim Heckman <weu_rznvyhfrarg@lnubb.pbz.invalid> wrote: > > >On 26Jul2006, baez@math.removethis.ucr.andthis.edu (John Baez) > >wrote in message <ea83ig$qmq$1@news.ks.uiuc.edu>: > > >> 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 onetoone correspondence > >> with the integers. There are much bigger *uncountable* ordinals  > >> at least if you believe you can wellorder uncountable sets. > > >? Is that last a reference to the WellOrdering Theorem (equivalent > >in ZFC to the Axiom of Choice)? Of course, you do need the WOT to > >prove that /every/ set can be wellordered, 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 wellorder > 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?
[...]
 Jim Heckman

