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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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 baez@math.removethis.ucr.andthis.edu Posts: 446 Registered: 12/13/04
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Jul 29, 2006 12:30 PM

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.

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