From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: Re: Axiom of Choice
Date: 5 Jul 2000 06:08:10 GMT
Newsgroups: sci.math
Summary: [missing]

|> >As for me, I feel uncomfortable whenever I have to use the Axiom of
|> >Choice, except for a countable set.

Join the club.   Incidentally, you probably would like to extend your comfort
zone from mere countable AC, to DC - the axiom of dependent choices.  
This is strictly stronger - it says not only can you choose one from each of 
a countable collection, but can do it even if each choice may have to depend
on the previous ones.  Most ctbl-AC-ers would have no difficulty with this
extension, which is far and away the most commonly used in analysis etc.


|> Most mathematicians would feel no discomfort whatever in using the
|> Axiom of Choice.

True; but quite a few do.

Incidentally, I was reading on the foundations-of-math mailing list recently,
that for quite some time theorems in journals would be stated with the proviso
"AC needed", when it was; and that this gradually faded away as authors (and
journals) gradually became inured to it.  I hadn't been aware of this before,
I'd thought that as soon as it was identified (Zermelo 1904?) it became
immediately accepted.   Can anyone enlighten us further on this matter?

BTW, Freidman strongly suggests that large cardinal axioms will gradually
assume the same status as their essential use becomes better known in
mainstream math, in particular, in "Boolean relation theory", his own
particular baby.  It will be interesting to see.  I hope I live so long!

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             Bill Taylor        W.Taylor@math.canterbury.ac.nz
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Truth decays into beauty, while beauty soon becomes merely charm. Charm ends up
as strangeness, and even that doesn't last - but up and down is for a lifetime.
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