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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 Jim Heckman Posts: 22 Registered: 7/13/06
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Jul 31, 2006 5:46 PM

On 29-Jul-2006, baez@math.removethis.ucr.andthis.edu (John Baez)
wrote in message <eag2eh\$bcq\$1@news.ks.uiuc.edu>:

> Jim Heckman <weu_rznvy-hfrarg@lnubb.pbz.invalid> wrote:
>

> >On 26-Jul-2006, 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 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?

[...]

--
Jim Heckman