🚀 go-pugleaf

In article <33375C6B.167EB0E7@owlnet.rice.edu>, Robert James Aguirre
<robba@owlnet.rice.edu> writes
>Martin Ward wrote:
>
> 
>> I was referring to real numbers (like pi - see the subject line) so
>> Jaime's crticism is valid.  I presume Ulrich was thinking of the natural
>> numbers, or possibly the rationals, or possibly rationals of the form
>> a/b where a,b are natural numbers with a<=31 and b<=12.  Any of these
>> sets can be well-ordered, and then Ulrich's argument "works" as Katy
>> says.
>> 
>> I like this argument, but of course it has a more subtle flaw...
>> --
>
>Hmm  I suppose the reals "can" be well ordered as well (or must I leave
>choice axioms alone...

Yes, you are right, given the Axiom of Choice any set can be well-
ordered.  However, we cannot assume that the "smallest" element in the
reals under an arbitrarily chosen well-ordering will be interesting,
since this is tantamount to assuming that every real is interesting.
(Given a well ordering of the reals we can easily construct another with
any desired smallest element).  Petitio principii.

This also applies to the rationals, of course - I was sloppy there, it
is too long since I really thought about this sort of thing.  However,
the natural numbers are well-ordered by the usual order <, and 1 is
interesting - this is what Ulrich's argument refers to - and the
rationals can be well-ordered by a bijection with the natural numbers
which maps interesting rationals on to 1,2,... and it is that that I was
thinking about above.  The reals cannot be well-ordered in any such
simple way.
-- 

Martin Ward

_________________________________________________________________________

"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
 means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
 different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".

"Through the Looking-Glass" - Lewis Carroll.
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