Re: Dedekind Cuts, Fundamental Sequences: why?

quasi wrote:

On Sat, 02 Jun 2007 17:57:21 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

What is the step of logic which leads one to seek an extention of the
rational numbers to the real numbers?

I understand the arguments given by Prickert and Görke for all of the
previous extentions. But when they go from the rationals to the reals,
they don't really present a formal equation in need of a solution as they
had for all of the prior extentions. It leave me to wonder what, exactly,
the criteria for success is.

Completeness.

You want limits to exist when there's information suggesting that the
limit should exist.

For example a bounded set should always have a least upper bound.

The rationals don't achieve this -- there are holes. The construction
of the reals is designed to patch all of the holes.

quasi

The doesn't seem very rigorous to me. How does one prove these holes exist?

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