Re: Is continuum completely filled up?

On Sat, 30 Dec 2006 14:42:52 +0900, ooo wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:en104u$oe7$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Here's a paradox: The reals are a complete ordered field. If you extend
the reals by including infinitesimals, the resulting extension field is no
longer complete. By adding things to a complete ordered field, we get an
incomplete ordered field.

I see. Thank your kind explanation. Cannot we extend that to complete field?

No. There is only one complete ordered field, up to isomorphism.

A difinition of infinity is ,therefor,ambiguous that a set of natural
number
contains only finit size of number ,but entire member of it is infinite.

How is that "ambiguous"? Is the color blue ambiguous because we can fill
a
blue basket with red objects?

And What do you label to that basket ? Can you know content by basket only?

Why do you ask?

We can refor to "round square " ,but it is meaningless.
That might not be good examle. But the expression of infinity using
countable number of simbols, may not be necesarily consistent.

Really? How so?

I concider that entire naturals is what we cannot comprehend.Infinity is
beyond our reach.

But we can reason about it.

Yes, We can reason about it as concept using sinbols. Therefore what we are
really dealing with with is countable numbers of symbols.

But we can reason about uncountable sets. The fact that there is only one
complete ordered field, up to isomorphism, is one example of such reasoning.


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... There is only one complete ordered field, up to isomorphism. ... We can reason about it as concept using sinbols. ... But we can reason about uncountable sets. ...
    (sci.math)
  • Re: Analysis with the smallest number...
    ... calculus when you can just accept the real number system for what it ... There are no infinitesimals in the real number system, ... ordered field that contains infinitesimals is most certainly not "within ... reals, by definition, are an Archimedean field. ...
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  • Re: how to list all of the real numbers
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  • Re: Is continuum completely filled up?
    ... incomplete ordered field. ... by saying that the field is incomplete. ... infinitesimals is bounded above, but has no least upper bound. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... the reals by including infinitesimals, the resulting extension field is no ... incomplete ordered field. ...
    (sci.math)
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