From: Jim Choate <ravage@EINSTEIN.ssz.com>
Date: Sun, 19 Jan 1997 20:56:01 -0800 (PST)
To: cypherpunks@toad.com
Subject: [Math Noise] (fwd)
Message-ID: <199701200456.UAA00404@toad.com>
MIME-Version: 1.0
Content-Type: text/plain



Forwarded message:

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

An arbitrary Real can be constructed from the Rationals. If we accept the
proposition, as posed apparently by you and others, of uncountable Reals then
your 'assumption' fails, otherwise the 'uncountable' members would be
countable.

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

A 'limit point' is not the same as 'equal to'. Arbitrarily close is not
equivalent, inherent in the definition of a limit point is the concept of
'little o' and 'big o', or worded differently our axiomatic definition of
infinity.

> 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.

We can, but there is no fundamental rule in mathematics that requires me to
ignore the small but distinct difference between the element in the original
set and the sequences used to approximate it. It is something that must be
agreed upon by accepting a particular axiomatic definition of infinity.

> 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.

The number of cuts are 1-to-1 with the Reals, they are not the Reals.
There is no way I can make a cut which is 3.1527, only 1-to-1 with the
number(s). Important distinction.


                                                  Jim Choate
                                                  CyberTects
                                                  ravage@ssz.com


"The best lack all conviction, while the worst are full of passionate
 intensity."

                                                     Yeats