
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted:
Jul 29, 2006 12:30 PM


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.
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":
http://www.henryflynt.org/studies_sci/mathsci.html

