From: Mike Duvos <mpd@netcom.com>
Date: Sun, 19 Jan 1997 19:55:29 -0800 (PST)
To: cypherpunks@toad.com
Subject: [Math Noise]
Message-ID: <199701200355.TAA29371@toad.com>
MIME-Version: 1.0
Content-Type: text/plain


Jim Choate <ravage@EINSTEIN.ssz.com> writes:

 >> One can construct the reals from the rationals quite easily
 >> using any of several well-known methods, such as
 >> equivalence classes of Cauchy sequences, or Dedikind cuts.

 > So you are saying that the Reals are a subset (ie can be
 > constructed from) of the Rationals?

Yes, the Reals can be constructed from the Rationals.  No, the
Reals are not a subset of the Rationals.

Fortunately, construction in mathematics is not simply limited to
the taking of subsets.  The Rationals can be constructed from the
Integers, for instance, by multiplication and the taking of
appropriate equivalence classes.

In fact, everything can be built out of two sets and the axioms
of Set Theory.

 > I can create a number which is not representable by the
 > ratio of two integers from two numbers which are
 > representable by ratios of two integers?

Er, no.  But you can create a number which is not representable
as a fraction as a limit point of very many fractions.

 > That's a nifty trick indeed, I am really impressed.

Thank-you. :)

 > Cauchy produced a test for testing convergence. I fail to
 > see the relevance here, but please expound...

Cauchy sequences are useful for adding limit points to a set of
things because their convergence criteria is very simple, and it
is conceptually easy to take all Cauchy sequences whose elements
come from a given set.

If we then consider equivalence classes of those Cauchy sequences
which converge to the same limit, and consider an element of the
original set to correspond to the class containing the sequence
all of whose members are that element, we can consider the
classes to form a "completion" of the original set by addition of
all its limit points.

Similarly, the Reals are the completion of the Rationals.

 > Dedekind Cut:

 > "Thus a nested sequence of rational intervals give rise to
 > a seperation of all rational numbers into three classes."

 > Just exactly where does this allow us to create Reals?

Finite ordered sets have maximum elements.  Bounded infinite
ordered sets have Least Upper Bounds, which may be a limit point
as opposed to being an actual member of the set.

Dedekind Cuts are a simple abstraction, often used to construct
the Reals from the Rationals in undergraduate calculus courses.
Conceptually, one makes a single "cut" in the set of Rationals,
dividing it into two parts, all of the members of one part being
greater than all of the members of the other.  The number of ways
of doing this correspond to the Reals.

--
     Mike Duvos         $    PGP 2.6 Public Key available     $
     mpd@netcom.com     $    via Finger.                      $