From: Mike Duvos <mpd@netcom.com>
Date: Mon, 20 Jan 1997 08:12:20 -0800 (PST)
To: cypherpunks@toad.com
Subject: Re: [Math Noise] (fwd)
Message-ID: <199701201612.IAA12442@toad.com>
MIME-Version: 1.0
Content-Type: text/plain


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

 > 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,

Again, we may speak of a representation for the Reals within a
formal system, although we may not speak of "The Representation
of X" for every single Real X.

We cannot assign to every Real a finite representation, but we
can talk about the infinite representation all Reals have within
a formal system without running out of space.

The inability of a formal system to talk about each Real
individually, or equivalently, that there are Real numbers which
are not finitely denumerable, does not mean that there are Reals
which do not have an representation as a non-ending sequence of
symbols.

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

 > such that there are uncountable numbers between m,n or p,q; etc.

Infinity does not have a predecessor, so it makes no sense to
count back from it a finite number of steps.

 > Clearly in contradiction with the base axioms of
 > mathematics as described by Euclid in defining a line.

This point completely escapes me.

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

Useful formal systems employ finite strings from some alphabet.
The set of all possible such strings is countable.  The set of
all sequences from the same alphabet is uncountable, but not
particularly useful for theorem-proving, at least in a finite
amount of time.

 > 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.

No.  Points do not have width.

 > 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.

In Axiomatic Set Theory, it is necessary to postulate (either
implicitly or explicitly) the existance of one infinite set. This
is an act of faith.  Whether any infinities really exist is a
matter for philosophy.

 > 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.

If one constructs the Ordinals, which are isomorphism classes of
well-ordered sets, and the Cardinals, which are equivalence
classes of equipotent sets, one will automatically end up with
all sorts of transfinite numbers.  We normally don't include
infinities when we build the rationals, the reals, or the complex
numbers, unless we need them for a particular application, such
as in using the extended real number line in defining measures.

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