Re: Dedekind Cuts, Fundamental Sequences: why?

Bob Kolker wrote:

Hatto von Aquitanien wrote:

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

Very simple. You want every set of numbers bounded from below to have a
greatest lower bound and every set of numbers bounded from above to have
a least upper bound. While rational numbers are dense in their ordering
they lack the closure of boundedness, hence real numbers are invented to
extend the rationals.



Bob Kolker

That appears to be the stated motivation of Pickert and Görke. Part of
their reasoning appears to rely on the expression which is incorrectly
transcribed on page 131. I have tried very hard to find the proper form of
that expression, but the only institution in my area which I know to be in
possession of these volumes appears to be incapable of making them
available for use. That institution, BTW, happens to be the University of
Maryland.

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Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... ordering they lack the closure of boundedness, ... the set of positive rationals whose squares are less than ... or equal to 2 is bounded above but has no least upper bound. ... or equal to 2 is bounded below but has no greatest lower bound. ...
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  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... ordering they lack the closure of boundedness, ... the set of positive rationals whose squares are less than ... or equal to 2 is bounded below but has no greatest lower bound. ... Pickert and Görke call fundamental sequences "Cantor sequences". ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... ordering they lack the closure of boundedness, ... invented to extend the rationals. ... or equal to 2 is bounded above but has no least upper bound. ... or equal to 2 is bounded below but has no greatest lower bound. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... You want every set of numbers bounded from below to have a greatest lower bound and every set of numbers bounded from above to have a least upper bound. ... While rational numbers are dense in their ordering they lack the closure of boundedness, hence real numbers are invented to extend the rationals. ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... ordering they lack the closure of boundedness, ... the set of positive rationals whose squares are less than ... or equal to 2 is bounded above but has no least upper bound. ... What do you mean by 'mapping' that ...
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