Re: The state-of-the-art in mathematics
From: robert j. kolker (nowhere_at_nowhere.com)
Date: 12/29/04
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Date: Wed, 29 Dec 2004 15:30:05 GMT
msherwood1@gmail.com wrote:
> Thanks.
>
> I definitely realized I might be wrong; your explanation was
> (considerably) more patient than his.
>
> On the previous gentleman's remark that Completeness can be constructed
> (i.e. doesn't have to be assumed), I'll definitely investigate. But I
> want to be clear about his..If above is so, why do we take (or why do
> many books take) as axiomatic "for every non-empty subset of the Reals
> that is bounded above, there is a Least Upper Bound". The more we can
> "construct", and accordingly, the less we have to take as "axiomatic",
> the better, no? And if we have to take this as axiomatic, isn't this a
> pretty big Leap of Faith?
The real numbers of constructed from the rationals so that Cauchy
sequences converge to a limit. Using this, one can show that every set
of numbers bounded above has a least upper bound. Left as an excercise.
Bob Kolker
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