Re: Attempts to Refute Cantor's Uncountability Proof?

On Tue, 11 Jul 2006 11:18:00 -0400, Hatto von Aquitanien wrote:
*** T. Winter wrote:

In article <BPqdnbM13PiEBy7ZnZ2dnUVZ_qydnZ2d@xxxxxxxxxxxxx> Hatto von
Aquitanien <abbot@xxxxxxxxxxxxxx> writes: ...
> > And we don't need to "suppose" we have a counted list of rational
> > numbers.
> > It's been done. See, for example,

<http://www.lacim.uqam.ca:16080/~plouffe/OEIS/citations/recounting.pdf>.

> No, you have an algorithm. I can produce an algorithm for generating
> "all" the "reals" iteratively. In a finite number of steps, it will
> only produce rationals, but I can show that it approaches the reals
> with every iteration.

But there will be no step that you generate an irrational. While in the
algorithm for the rationals, for each rational there is a finite step that
it is generated.

What I'm trying to get at is the essential difference between these two
concepts. One says I can visit each rational number and assign a serial
number to it, ad infinitum.

No, nothing is stated about what you can do. All that is stated is that
there exists a mapping f: N -> Q, and that this mapping is bijective.

The mapping is basically a set G (for *graph*) of ordered pairs from NxQ
with the property that for each n in N, there is a unique q in Q such
that the pair (n,q) belongs to G. If we can describe the set G, so that
we know exactly which ordered pairs are members and which are not, then
we have our function f.

Of course that really says _IF_ we could
actually perform that infinite number of steps... At some point we make a
transition from "if we could" do "we can".

The words "we" and "can" do not appear anywhere in the definition of a
function.

We do something similar to my
algorithm when defining or "proving" integration as a limit of a Riemann
sum. The reason we reject my algorithm is because we "know" it will never
exactly produce a number with an infinite decimal expression in a finite
number of iterations. We can show that through induction as I did
previously. But I could say the same thing about Riemann sums.

The difference is that the Riemann integral is defined by a type of limit
process, while the definition of surjectivity of a function has nothing
whatever to do with limits. In fact, surjectivity does not even require
a topology, which is a prerequisite for speaking of limits.

There appears to be an essential distinction between these two situations
which makes one acceptable, and the other one unacceptable.

It's acceptable to speak of limits when dealing with the Riemann
integral, because the definition involves a limit. It's not acceptable
to introduce limits when discussing surjectivity, because limits are
irrelevant to the definition of surjectivity.

With the case
of Riemann sums, we have a bound index variable to enumerate the individual
rectangles we are summing over. If I try to draw an analogy between the
index of each rectangle of a Riemann sum and the index I'm generating with
my algorithm, something has to be different. But I'm not sure I know what
that something is.

Does this help?


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

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