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.

This is the motivation stated by Pickert and Görke, but it is not clear to
me why it matters. Nor is it completely clear what it means to say that
the field of rational numbers does not exhibit the closure property of
boundedness.

For every rational number I can very easily divide the rational numbers into
two disjoint sets by asserting that every rational number greater than the
selected number is a member of the set whose lower bound is the selected
number. Likewise for the symmetrically opposite case. I then arbitrarily
chose one side of the bifurcation to include the chosen rational number.

Every definition I have consulted for supremum and infimum begins with the
real numbers. So to tell me that the reason we need to extent the rational
numbers to the real numbers is so that the domain of numbers has suprema
and infima assumes the real numbers to be defined already.

http://www.cuyamaca.edu/bruce.thompson/Fallacies/circ_justification.asp
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Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... Bob Kolker wrote: ... greatest lower bound and every set of numbers bounded from above to have ... 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. ... or equal to 2 is bounded below but has no greatest lower bound. ...
    (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 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)