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From: "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
Newsgroups: sci.math
Subject: Re: Foundations of real numbers
Date: Sat, 13 Oct 2012 13:46:53 +1100
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"Shmuel (Seymour J.)Metz"  wrote in message 
news:5078204a$24$fuzhry+tra$mr2ice@news.patriot.net...

In <k585os$h3...@news.albasani.net>, on 10/12/2012
   at 03:19 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:

>Because in order to define a Real number you have define a mapping
>from N to Rationals.

You're restating the phrase instead of explaining why it is a problem.

>Lots of situations arise where there is no obvious mapping.

1. You haven't explained why you need an obvious mapping

2. You haven't given an example where it it is a problem.

>The LUB property makes it easy to say that the sequence pi, pi-3,
>pi-3.14,  pi-3.141, ... defines a unique Real;

Since each of those terms is irrational, it is not easy.
__________________________________________
Yes it is. It is a set of Reals with an upper bound of 4. It must therefore 
have a LUB. How hard was that?


Since the
sequence is a Cauchy sequence, you have the obvious mapping that you
say you need.

________________________________________________
To use the Cauchy sequence to construct R, the terms must all be rational or 
else the construction is circular. "The real numbers are complete under the 
metric induced by the usual absolute value, and one of the standard 
constructions of the real numbers involves Cauchy sequences of rational 
numbers.", from http://en.wikipedia.org/wiki/Cauchy_sequence . You will note 
the terms in pi, pi-3, pi-3.14 ... are not rational. So we do not have a 
mapping from N to Q as you claim.




-- 
Shmuel (Seymour J.) Metz, SysProg and JOAT  <http://patriot.net/~shmuel>

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