Re: Cantor's diagonal proof wrong?

From: Curt Welch (curt_at_kcwc.com)
Date: 11/18/04 

Date: 18 Nov 2004 22:07:40 GMT

stephen@nomail.com wrote:
> Curt Welch <curt@kcwc.com> wrote:
> : stephen@nomail.com wrote:
> :> 1/5 is not a procedure. It is a number. .2 is also a number.
> :> 1/5 and .2 are different names for the same thing. What procedure
> :> is .2?
>
> : Counting is a procedure. When I ask a 6 year old what comes after 1
> : and he says "two", he has correctly performed the successor procedure.
> : We train people to correctly apply the successor procedure so they can
> : always produce the next natural number given a starting number.
>
> : In this regard, it's reasonable to say that 4 represents x x x x
> : iterations of the counting procedure.
>
> : I'm not saying it is wrong to call 1/5 a number. I'm saying that
> : behind those words is a hidden procedure which is the true source of
> : the meaning of those words, but yet, we are taught to objectify the
> : reference of the sign and pretend it's an object, and not a procedure.
> : But behind all that objectification done in the language, is a
> : underlying foundation of procedures.
>
> : .2 represents the results of dividing 1 by 5. Division is the
> : procedure for finding the number when multiplied by 5, will produce 1.
>
> .2 also represents the result of adding .1 and .1. It also represents
> the result of taking the square root of .04. It also represents
> the result of summing (2^i)/10 for i=0 to infinity. I can come up
> with an infinite number of procedures that produce .2. Claiming
> that .2 is one and only one of those procedures seems silly.
> If you claim that it is all of those procedures than you have
> to claim that it is the result of both finite and infinite procedures,
> which will be true of all numbers.
>
> IMO, claiming that 1/5 is a finite procedure but 1/9 is not
> makes no sense. There is nothing infinite about dividing
> something by 9.

Yes, you what you say is true.

The point I've been making is not an issue about which procedures terminate
and which doesn't. The point I've been making is the meaning behind all of
it is procedural in nature. That is the foundation which math grows out of.
And sometimes, the reference we make are to procedures which do not
terminate.

The fact that we make reference to procedures that do not terminate however
does not prevent us from talking in intelligent ways about the nature of
the procedure. We know that an infinite series will approach a limit and
we don't have to perform the infinite procedure to know that. If the limit
it approaches is 2.0, then we know that we can substitute the value 2.0 in
the equation where the infinite series was, and get the right answer - and
at the same time, change the procedure from an infinite one, to a finite
one (counting to two).

I have no problem, or confusion, about how we make good use of all the talk
about procedures that loop forever.

However, when you start talking about constructing a 1 to 1 mapping between
to procedures which loop forever, and tell me that it can't be done, I get
lost.

To talk about a 1 to 1 mapping between two finite sized sets of objects is
simple and clear.

To talk about 1 to 1 mapping between a finite set, and a procedure that
never terminates is also clear - it can't be done.

To talk about a 1 to 1 mapping between to infinite procedures, makes no
logical sense to me. It's always possible to create the mapping, because
no matter how many objects are generated by each procedure, you can always
match them up as they are generated (this idea I'm beginning to see seems
to be related to the axiom of choice).

So, if the meaning behind all math concepts is ultimately derived from
procedures, and infinite procedures can always be 1 to 1 mapped, how is it
we are able to find the words to prove that some infinite procedures are
not 1 to 1 mapped with all other infinite procedures?

I just don't know how the words were found to prove that yet. But there
are a lot of words used in the formal foundation of mathematics that
everyone says does prove it. So I'm just curious how mathematicians did
what seems to be the impossible.

For example, if look at Cantor's diagonal argument with the belief that all
references to infinity are references to processes that don't terminate,
you see something very different than what is normally seen in the proof.

The proof starts with the assumption that there is a mapping. This would
imply that there is a procedure able to sequentially generate all the
reals, one at a time. Because a single real might in itself be represented
by an infinite procedure (to generate all the digits of PI for example),
you can think of the problem as a procedure for generating an infinite set
of procedures which in turn, generate each possible real.

Now the argument says you need another procedure running in parallel to
generate the diagonal. This procedure takes each real generated by the
first procedure, and grabs the Nth digit from it. Since for each real,
it's grapping a finite digit, it doesn't even need to let the procedure
continue generating all the digits. So that's not a problem.

Now it takes the Nth digit, and changes it. It continues in this fashion,
for as long as the real generating processes produces reals to work with.

We know for a fact that this diagonal number being generated will be
different from every past real generated. But, at any time T, we have only
built N digits of the diagonal so far, and we have no way of knowing if the
partial real diagonal we have built so far, is located further down in the
table somewhere.

This procedure for generating reals, and generating the diagonal never
ends. It can't end because the procedure for generating reals can never
end.

But yet, we now claim, that the existence of the above never ending
procedure, is proof that there must be some real that the first procedure
will never produce.

I hope all of you can see that when presented like this, (as never ending
procedures) the proof is no proof at all. If not, I'll make the same style
argument for natural numbers...

Start with the counting procedure which produces all the natural numbers
starting with one (or zero). Now, generate a number which is guaranteed
not to match any of the numbers produced so far. We will do that by
picking N+1 where N is the highest number produced so far. By obvious
inspection, N+1 is a number which is different from every number produced
so far. Or, we can always generate a number which is not (yet) in the
table.

So does this prove that our procedure for producing natural numbers will
not be able to produce all the natural numbers? Of course not.

So, this is the heart of my confusion about the diagonal argument. It only
seems to make sense if you ignore the fact that all math is actually built
on the physical world notion of procedures, and procedures take time to
produce each new result, which means procedures which produce an infinite
number of results, can never terminate.

When, and why, was it ok for math to ignore its foundation in procedures?

What words can we point to that say, here is where mathematicians decided
to ignore the laws of nature that controls all procedures and make up their
own rules?

If that is what happened, I'd like to find those words and understand how
they violated the laws of nature.

OR, is it possible that a procedure based foundation of math can explain
why it's ok to say that sets with a cardinality greater than |N| exists?
If so, I'd like to see how that works, because the diagonal argument alone
doesn't seem to hold water when cast in a procedure based foundation.

Without a better understand of all the complex language used to justify the
foundation of math which leads to the hierarchy of cardinal sets, I can't
know for sure what happened. However, my best bet at the moment is that
mathematics, over the centuries, lost track of its roots, and simply choose
to define a few laws of nature out of existence. The most likely law they
choose to ignore is that nothing can be created without a physical agent
doing work and taking time. That is, they assumed that they have machines
which could follow these logical procedures (definitions/axioms), and run
at infinite speed to produce an infinite amount of results in zero time.
And when that happened, it became valid to use logic such as NOT FOR ALL,
on infinite sets, as if it were a finite sized set. Because in the real
world, NOT FOR ALL, is an invalid argument for an unspecified procedure in
a proof by contradiction argument.

But, like I have said, to uncover this mystery, I have to study the
language of math more. 

-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
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