[Math Noise] (fwd)

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

Jim Choate allegedly said:
> 
> 
> 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

No, that's not what he is saying.  What you have written does not represent 
a *specific* number.  He is saying that IF you have a particular scheme for 
representing *specific* numbers, you can only represent countably many 
-- for any given scheme, there are numbers you can't represent. 

To put it another way a scheme that says "you can represent numbers 
as half infinite strings of digits with a single period somewhere" 
doesn't actually *specify* any numbers.  A scheme that says "start 
with the number 1 and increment it 400 times" actually specifies a 
number. 

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

To be a formal system of the type required, you would also have to 
specify deterministic rules that could generate the "Ai" values.  The 
key distinction is between "expressed by" and "generated by".


-- 
Kent Crispin				"No reason to get excited",
kent@songbird.com,kc@llnl.gov		the thief he kindly spoke...
PGP fingerprint:   5A 16 DA 04 31 33 40 1E  87 DA 29 02 97 A3 46 2F

Jim Choate wrote:
> 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

All reals are equivalent to sequences of digits, but there are reals such 
that there is no algorithm to generate their digits.

It happens because there are "more" real numbers than algorithms.

> > 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.

I suggest the following mental exercise. FORGET FOR A MOMENT ABOUT 
REAL NUMBERS. Let's deal with mummies: 

DEFINITION: I define a mummy as possibly infinite sequence of
characters, separated by one dot, such that only characters abcdefghij
are allowed. Also, mummies that are represented by finite sequences of
characters are by this definition equivalent to mummies that end
with an infinite sequence of letters "a". END DEFINITION.

Examples: 

dce.abdefhaabdaaa
ae.cacacacacacaca... 

and so on.

Obviously, some of the mummies, such as c.cccccc... (with an ininite 
sequence of "c") CAN be generated by algorithms.

The interesting fact, that i will prove below, is that some of
them cannot be generated by any algorithm.

THEOREM: The set of mummies is more than countable
PROOF: if it is countable, we can construct a mummy that is not counted.
it is easy.

THEOREM: there are mummies such that there is no algorithm that
can print them. 
PROOF: the set of mummies is more than countable, the set of algorithms 
is countable, therefore there is no way to construct a one-to-one
correspondence between mummies and algorithms.

Do you agree?

Now let's back to the original problem of real numbers: the only
difference between mummies and real numbers is that digits 0123456789
are replaced by characters abcdefghij. 

Not a whole lot of difference, so everything that applies to mummies
applies to real numbers.

	- Igor.

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.                      $