Re: Attempts to Refute Cantor's Uncountability Proof?

Dave L. Renfro wrote:

Hatto von Aquitanien wrote (to kunzmilan):

So your method of generating all the members of an infinite
set failed.

Try this: Since I am told that it is meaningful to talk about
generation ad infinitum, why not start by something simple.
I have an algorithm which generates n.0, n.1,..., n.9, on the
first pass. On the second, it generates n.01, n.02, ..., n.09,
n.10, n.11, ..., n.19...,n.99. Where n is the non-negative
integer being visited. First visit 0, and make one pass,
then visit 1 and make one pass, return to 0 and make a second
pass, then to 1 and make a second pass, then visit two, etc....
Eventually, you will construct every number representable in
decimal notation.

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.

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.

Every number I generate increments a counter by 1.

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.

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.

Besides, even if there were a "countable infinity" among the
positive integers (and there isn't), you'd then have to tell us
what corresponds to the infinite decimal for 1/6, and then what
corresponds to the infinite decimal for 1/9, and for many other
(infinitely many other!) positive rational numbers as well.

I could throw those in the loop as well. I apply a predicate which
determines if there is a finite decimal representation for the rational
number, if so I ignore it, if not I add the infinite decimal
representation. That kind of messes up my original goal of saturating the
decimal places sequentially. OTOH, I just accepted an infinite chunk of
data. That deserves closer examination. It smells a whole lot like
circulus in demonstrando.


--
Nil conscire sibi
.

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