Re: Cantor's diagonal proof wrong?

From: *** T. Winter (***.Winter_at_cwi.nl)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 04:24:19 GMT

In article <20041120000433.576$gt@newsreader.com> curt@kcwc.com (Curt Welch) writes:
> "*** T. Winter" <***.Winter@cwi.nl> wrote:
> > In article <20041119191132.166$3I@newsreader.com> curt@kcwc.com (Curt
> > Welch) writes: ...
> > > I can write Sum(n=1 oo | 3/(10^n)) and I can replace those words with
> > > 1/3 in any equation, and not change the value, the meaning, or the
> > > truth, of that equation.
> > ...
> > > So, for any concept which translates to a processes which never ends,
> > > we know that the process can't actually exist, but it can still be
> > > useful to talk, and think, as if it did.
> >
> > But in mathematics it is *not* thought that the actual process of adding
> > infinitely many numbers together does exist.
>
> Yes, and I have no issue with infinite series. I never said I did. I just
> said the infinite processes doesn't actually exist because it doesn't and
> can't complete (and that confused some people as to what my point was
> probably because they haven't read all my posts in this rather out of
> control thread). We simply use the logic of limits to determine, and
> define, what numbers each series is equivalent to. And that's great,
> valid, and useful.

But mathematics does not consider infinite processes either (not entirely
true, see the halting problem). Your statement above: "determine", assumes
the existence of infinite addition. There is no such thing. The value of
the infinite series is not "determined", it is "defined". There is no
infinite process involved. You are confused by notation. What appears
as an infinite sum is not that, because such things do not exist in
mathematics; it is a limit.

> If you were given a description of the process which mapped the natural
> numbers to the reals, say by a function such as f(n) = r, then you can
> analyze what it will do without having to actually run the processes to
> completion - as is commonly done with infinite series and which is the
> foundation of calculus.

The only information f(n) = r is giving is that for every natural n, f(n)
will give a real in finite time. There is no reason to "run it to
completion", whatever that means.

> But in Cantor's argument, the actual mapping function is not given. It's
> only assumed to exist.

That is quite normal in a proof by contradicion.

> Then they argue that the diagonal anti-value can be
> constructed for any mapping function provided - which is still a valid
> idea. But then they make a conclusion that is invalid. They assume that
> since the anti-diagonal value being constructed doesn't match any single
> row, that it's valid to say that it doesn't match all the rows.

Eh? In what way is it possible that a thing that doesn't match any single
entry from a list, that it does match at least one entry from the list?
I would think that if something does not match any member of some set
it is not a member of that set, but you appear to think different.

> And as
> valid and as logical as that sounds as that sounds, it's not at all valid
> when you are constructing infinite sized real values in a infinite sized
> table. This is because it's impossible to construct the entire
> anti-diagonal value, and any part you have constructed, can always be part
> of a number that shows up further down in the table.

This is a misconception. Of course can every part you have constructed
be part of a number that shows up further down. But this is assuming that
you are constructing the number digit by digit from the first onwards. But
there is no need for such a sequential construction. The first step is
the assumption of a mapping f(n) = r, i.e. giving an arbitrary natural
number n, it will give a real number. The next step is the Cantor step.
Given a mapping f, there is a precise way to define a (single) number c(f),
such that that number is not in the image of the mapping. That number can
be defined by the following rules:
1. The n-th digit of c(f) is 4 if the n-th digit of f(n) = 5.
2. The n-th digit of c(f) is 5, otherwise.
No infinite process involved.

> But if you
> instead, choose to treat the construction of the anti-diagonal as something
> that can complete, then the logical argument that it exists in no row of
> the table is valid, and you get the results which have been well accepted
> for 100 years.

I take exception to the "construction that can complete". What does
"can complete" mean in this context? When is it possible to "complete
the construction" of, say, sqrt(2)? Does pi exist? Does e exist? In
the above, I would say that the construction of that number is complete,
just like I would say that defining pi as the ratio between the
circumference and the diameter of a circle completes the construction
of pi. That you can not write out all the base-10 digits in finite
time is no problem; you can do so in base-pi digits (but in that case
you would have a problem with 2). 

-- 
*** t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~***/
Quantcast