Re: Cantor Confusion

On 12/7/2006 5:24 AM, Virgil wrote:
In article <4576FD8D.60408@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:


It started with the question how to deal with the nil in case of
splitting IR into IR+ and IR-. I got as many different and definitve
answers as there are possibilities. I expected that there is only one
correct answer, and I found a reasoning that compellingly yields just
one answer in case of rationals and a different one in case of reals.

It might depend on the purpose of performing the split. I can think of
several ways to deal with it, at least one of which should be
satisfactory for any given purpose.

If, for example, one is trying to create Dedekind cuts, about the only
requirement is that if one puts 0 in one of IR+ or IR-, one goes the
same way with the boundary rational at every rational cut.

dedekind himself let the question open whether his cut belongs to the
left or the right side. He was also lacking insight.

.

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