Re: Dedekind Cuts, Fundamental Sequences: why?

In article <TKidnbd-d8d7Rf7bnZ2dnUVZ_jmdnZ2d@xxxxxxxxxxxxx>,
Hatto von Aquitanien <abbot@xxxxxxxxxxxxxx> wrote:

Virgil wrote:

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.

Isn't that just a fancy way of saying x^2=2, x>0 has no solutions in the
rational numbers? The example helps me understand how a set of rational
numbers could have no least upper bound in the rational number.

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

If I tried something creative such as using the set of all upper bounds as a
means of defining the least upper bound I guess I'd end up with Dedekind
cuts.

Yup!

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.

In this case, there seems to be an algebraic justification.

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.

In this sense, incompleteness seems to mean that mappings similar to
{x:x^2=2, x>0} are not endomorphic.

In my vocabulary, {x:x^2=2, x>0} qualifies as a set, but not as a
mapping. What do you mean by 'mapping' that {x:x^2=2, x>0} would be one?

That begs the question of what exactly
is meant by "mappings similar to {x:x^2=2, x>0}".

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.

I am convinced this is correct, but there is something about it that just
makes it hurt to think about it too hard.

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.

The definitions I consulted presuppose the existence of the real numbers.

The definitions of 'supremum' and 'infimum' that I am aware of, only
rely on the set in question being ordered. Any other properties are
extraneous. What definitions do you find that demand anything other than
order?

I note, however, that Pickert and Görke introduce 'fin M' corresponding with
every non-empty set M of rational numbers bounded from above. In a
footnote they tell us that fin M initially has no further meaning. It
eventually becomes the definition of a real number. IB1 §4.3 page 134

http://baldur.globalsymmetry.com/open-source/org/sth/math/behnke-et-al/vol-1.d
jvu

One aspect which makes their development difficult to follow is that they
introduce three different means of arriving at the real numbers.

It transpires, though often at great length, that the differing paths
through which one can go to develop those numbers lead to isomorphic
results. So follow one path at a time till it is clear before worrying
about others.
.

Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... That still begs the question. ... boundedness. ... S does not have a rational least upper bound. ... I explained exactly why the rationals are not complete - ...
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  • 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. ...
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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 above but has no least upper bound. ... means of defining the least upper bound I guess I'd end up with Dedekind ...
    (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. ...
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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". ...
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