Re: Attempts to Refute Cantor's Uncountability Proof?
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: 8 Jul 2006 20:29:18 -0700
Dave L. Renfro wrote:
At what point in your list will 1/3 be reached? A rough
estimate would be acceptable.
Hatto von Aquitanien wrote:
When I reach countable infinity.
Dave L. Renfro wrote:
The context is a list that has a first element, a second
element, a third element, and so on, for each positive
integer. There is no "countable infinite" position on
such a list.
Hatto von Aquitanien wrote:
Every number I generate increments a counter by 1.
Fine. This means the counter will show '1', '2', '3', etc.
But at no point will the counter show "countable infinite".
Dave L. Renfro wrote:
To show that the positive rationals have the same cardinality
as the positive integers, you need to assign (in a unique way)
a certain positive rational number to '1', a certain positive
rational number to '2', a certain positive rational number
to '3', and so on, in such a manner that every positive
rational number is used up. There is no "countable infinity"
among the numbers '1', '2', '3', etc. -- you have to stay
with the numbers '1', '2', '3', etc.
Hatto von Aquitanien wrote:
And never mind that you are using integers up a lot faster than
you are counting them. That simple fact right there is enough
to make the whole proposition hard to accept. That really is
the crux of the argument that methods created for dealing with
finite sets are being abused by applying them to infinite sets.
If I ask at any given point during this enumeration process
what's the difference between the number of integers already
counted, and the number consumed, the latter explodes. One
typically considers such numerical behavior to be divergence.
I guess the argument Cantor will give is that at any given
point in the process, there is a bijective map, and then apply
induction.
What about the fact that I can list (in the manner we're
talking about) all the positive integers using the positive
even integers? In what follows, the counting is being done
by the positive even integers (left of the arrows) and the
numbers being counted are the positive integers to the
right of the arrows.
2-->1 4-->2 6-->3 8-->4 10-->5 12-->6 14-->7 . . .
In this situation, I'm using up the even integers that I'm
"counting with" much faster than the even integers that are
among the numbers I'm counting. The (arithmetic) difference
between the number of even integers counted vs. the number
of even integers used up increases without bound (or "explodes",
as you say).
Instead of even integers, if I use powers of 2 to count with,
I can make the gaps increase fast enough so that we actually
get the *ratios* between the number of those integers that I'm
using to count with (i.e. the powers of 2) vs. the number of
powers of 2 that get used up to increase without bound:
2-->1 4-->2 8-->3 16-->4 32-->5 64-->6 128-->7 . . .
These are 1-1 correspondences, by the way. In the first case,
each positive even integer n is matched with exactly one positive
integer, namely n/2, and each positive integer m is matched with
exactly one positive even integer, namely 2m. In the second case,
each positive integer power of 2 is matched with exactly one
positive integer, namely the exponent of 2 associated with the
power of 2, and each positive integer m is matched with exactly
one positive integer power of 2, namely 2^m.
With the right approach, one can use the positive integers
"to count" the positive rational numbers. Although Cantor
is credited with this, I don't think it is considered all that
significant, and in fact I believe that Dedekind (who Cantor
was in correspondence with during 1873-74 when Cantor was
taking his first steps into this area) independently came
up with such an approach. Cantor's importance comes from
the fact that: (1) he first considered in a systematic way
what the consequences might be for comparing infinite sets
in this way (by 1-1 correspondences); (2) he managed
to come up with a proof that there exists an infinite set
that can't be "counted" by using the positive integers
(namely, the real numbers); (3) he created an elaborate
set of mathematical tools that proved very useful for
solving problems, and establishing connections, in a wide
variety of mathematical fields.
Dave L. Renfro
.
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