From: Jim Choate <ravage@EINSTEIN.ssz.com>
Date: Sun, 19 Jan 1997 20:57:46 -0800 (PST)
To: cypherpunks@toad.com
Subject: [Math Noise] (fwd)
Message-ID: <199701200457.UAA00416@toad.com>
MIME-Version: 1.0
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Forwarded message:

> Only countably many real numbers, or members of any uncountable
> set, are denumerable. It is the property of being uncountable,
> rather than of being real or complex, which is important here.

In short you are saying there are Reals which can not be expressed in the
format:

     AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1

where m and n -> infinity. Another way of saying this is that there
are Reals for which membership in a set of Reals, because they are
uncountable and therefore unrepresentable, is not possible. From a set
perspective this means,

R = [[-infinity, ..., m] [n, ..., 0] [0, ..., p] [q, ..., +infinity]]

such that there are uncountable numbers between m,n or p,q; etc.
 
Clearly in contradiction with the base axioms of mathematics as described by
Euclid in defining a line.

> In general, only countably many members of any uncountable set
> can be precisely specified within any formal system, given names
> comprised of strings of symbols, or other similar things.

And I contend that ANY number which is Real can be expressed by the decimal
expansion above. Which clearly qualifies as a formal system.

There are three ways of looking at the total of mathematics,

symbolic

set theory (Bourbaki School)

geometric (Euclidian)

While it is true the 3 are equivalent, the choice of approach does have a
relevance on how difficult, if even tractible, the proof is for any particular
concept.

> It is often convenient, such as when drawing contour maps, to
> consider the complex numbers to be in 1-1 correspondence with the
> points of the plane.  However, I wouldn't necessarily consider
> regions of the complex plane to have "area" in the Euclidian
> sense.

The Euclidian plane IS a complex plane. The -j or -i used in Complex
symboligy simply means 'rotate the second axis of measure 90 degree
counter-clockwise (by agreement) to the axis of the first measure'. If it
requires 2 or more numbers (integer or real) taken as a set to represent the
quantity it is a Complex. Regions of a Complex plane are directly
comparable (1-to-1) with the concept of 'area' on the Euclidian plane.

> We can't physically draw a line segment to arbitrary high
> precision.

Irrelevant. If the hole in my approach is that I can't draw a line of
arbitrary precision in practice then your own 'uncountable numbers' argument
falls for the same reason because if it is truly uncountable you can't point
to it on a number line and say "there is an uncountable". This discussion is
one of principles, not one of practice.

> We can conceive of the notion of line segments being
> in 1-1 correspondence with the reals, but we can specify at most
> countably many "finitely denumerable" line segments if we wish to
> discuss their lengths individually.

To say there are Reals for which there is no linear representation is the
same as saying there are lengths which can't be measured. Now since a line
is nothing but a set of points, which don't have size, but only position
this obviously holds no water, unless you are saying it is not possible to
place two points arbitrarily close together, which would imply that points
have some sort of width, clearly against the definition of a point.

This all goes back to what I said in a earlier post, the problem comes from
our axiomatic (ie taken on faith, unprovable) use of infinity. Without a
clear and precise dilineation of those axioms prior to the proof such
conclusions are worthless. Several of you have said "infinity is not a
number", this is an axiom. Change the axioms and the whole structure
changes. I am simply saying that perhaps we should look at the "infinity is
not a number" axiom, much as geometers look at Euclids Fifth Postulate.
There is nothing inherent in nature that prefers one axiomatic expression
of infinity over the other.

By changing our axiomatic definition of infinity we reduce the sets we have
to work with from [Integer, Irrational, Real, Complex] to [Integer, Real,
Complex]. Now, whether it is worth the trouble is at this time unanswerable
because nobody has ever done the research.


                                                   Jim Choate
                                                   CyberTects
                                                   ravage@ssz.com


"The laws of mathematics, as far as they refer to reality, are not certain,
and as far as they are certain, do not refer to reality."

                                                  Albert Einstein