Re: Dedekind Cuts, Fundamental Sequences: why?

On 2007-06-05, in sci.math, Hatto von Aquitanien wrote:
That tells me how they are useful, not why. But what I really meant by my
previous comment is that he was begging the question as to why fundamental
sequences or Dedekind cuts are useful in defining completeness. And I
still do not know if there was a clearly defined objective which can
subsequently be verified as being accomplished.

They provide a concrete way of specifying a complete ordered field with a
countable dense subset, reducing talk about reals to talk about naturals and
sets of naturals. Such a reduction is of importance in that, provided we
pretty much agree what sort of principles and modes of reasoning are
acceptable when dealing with naturals and sets thereof, we get a clearer
view of what sort of reasoning about reals, functions of reals, and so on,
is acceptable. This might or might not be something that is of interest to
you, but historically it was one of the main reasons for the "arithmetical"
construction of the reals.

As to your other point, it is not really so much about having solutions to
equations -- if that's all one cares about there is no need for completeness
-- as it is about limits and that sort of stuff from analysis.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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