Re: Dedekind Cuts, Fundamental Sequences: why?

In article <Vf-dnV0lgNg_4v7bnZ2dnUVZ_qmpnZ2d@xxxxxxxxxxxxx>,
Hatto von Aquitanien <abbot@xxxxxxxxxxxxxx> wrote:

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.

For example, the set of positive rationals whose squares are less than
or equal to 2 is bounded above but has no least upper bound.

Similarly, the set of positive rationals whose squares are greater than
or equal to 2 is bounded below but has no greatest lower bound.

So that there is a "hole" between those two sets.

For geometric reasons (having to do with the diagonals of squares), we
want that hole to be filled by something. But it cannot be filled by a
rational number.

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.

I am more familiar with this property being called lack of
"completeness" than lack of closure. For me, "closure" is usually
reserved for more obviously topological issues.

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.

There are, as noted above, other ways of partitioning in which there is
no rational number to serve as the divider.

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.

Not necessarily defined already, but capable of being defined in such a
way as to provide the missing "completeness" property.
.

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)