Re: Cantor
- From: jdolan@xxxxxxxxxxxxxxxxxxxxxxxx (James Dolan)
- Date: Wed, 5 Oct 2005 12:17:46 +0000 (UTC)
in article <qpmdnais27tdk97ervn-rw@xxxxxxxxxxx>,
pedro <someone@xxxxxxxxxxxxx> wrote:
|I think I'm getting sidetracked by giving poor examples. Can't see
|the forest for the trees, and all that... Let me ask it one more way,
|if I may.
|
|It's obviously impossible to actually present a "complete" list of
|reals prior to initiating the diagonalization method as the list is
|infinitely long. Rather, the diagonalization method is a thought
|experiment that says, in effect, "no matter how complete one makes
|the list, one can use this typographical method to find with
|certainty a number not on the list".
|
|But I can make the same conceptual argument about the natural numbers
|(or non-negative integers, if you prefer). In my example below, SSSS0
|differs in the first position from 0, differs in the second position
|from S0, etc..., thereby giving a number that differs from all others
|on the list and demonstrating an absurdity: that the natural numbers
|are not enumerable. (The only new assumption / constraint I've
|placed on this example is that the list be ordered. That doesn't seem
|to be an unreasonable thing to do with the natural numbers.)
|
|So I'm obviously missing something fundamental about the whole
|diagonalization thing. Any idea what it is?
it's hard to tell for sure, but it looks to me like the main thing
that you're not grasping is that there aren't enough _independent_
degrees of freedom involved in the act of choosing a natural number
for your stratagy of "one modification on your number per each number
that you need to establish it as different from" to succeed. you're
trying to construct a natural number n that's different from each of
the numbers 0,1,2,...; you make some decision about n to make it
different from 0, and then another decision to make it different from
1, and then you try to keep going like that forever, but it doesn't
work because the decisions that you're making aren't independent from
each other; there's a constraint that says that when all of your
decisions are combined together they can't be inconsistent (for
example by requiring n to do something impossible such as be a natural
number greater than all of the natural numbers in the world).
in contrast, there _are_ enough _completely independent_ degrees of
freedom involved in the act of choosing a real number for you to
construct a single real number r that establishes itself as different
from each real number in some countable list. by definition, when you
construct a real number r you're allowed to decide what each digit in
r should be, completely independently from all of the other digits in
r. (there's a minor glitch in the sentence that i just wrote that you
can fix for yourself, or someone can fix it for you.) the division of
labor is then that the nth digit in r does the work of establishing r
as different from the nth real number in some countable list. each
digit is free to do its work in complete seclusion from worry about
what all of the other digits are doing, because of the nature of real
numbers: that digits in them can essentially be chosen completely
independently from all of the other digits.
--
[e-mail address jdolan@xxxxxxxxxxxx]
.
- References:
- Cantor
- From: Pedro
- Cantor
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