A new point of view on Cantor's diagonalization arguments

matt grime

It would be nice if you told us what the uncertainty and redundancy of a number is. So that we know what we MUST be aware of.

Organic

The answer is here:

http://www.geocities.com/complementarytheory/count.pdf

matt grime

Ah, of course! The answer to my question is in another pdf! This one doesn't even mention the words uncertainty and redundancy. In what way is that an answer?

Do you mean to imply that numbers are uncertain because if we have set of identical objects we cannot distinguish between them? I don't see why that makes numbers uncertain. I see why it makes identifying identical objects impossible, but that has nothing to do with quantity. Suppose I just had one bead. I blink, I see an identical bead in the same place. is it the same bead? I do that for more than 1 bead.. what has quantity got to do with it?

Organic

From your response it looks that you did not read the pdf file.

Please reply if you have some technical problems to open the pdf.

Thank you,

Organic

matt grime

I opened the pdf link you gave and didn't see the word uncertainty mentioned once, please tell me which line it is on. Or, as the pdf is only a page long, try stating in plain simple English what you mean the the uncertainty of a number, or its redundancy. It's another simple request. Actually it's the same simple request isn't it? A simple paragraph starting:

The redundancy of a number is...

or perhaps it ought to start

A number is redundant if...


or even,

given a number n, and some object related to n, then the object is redundant if...

Organic

Dear Matt,

You are going too far.

This pdf is not a technical paper, but a simple test that answer to the question:

what are the minimal conditions that give us the ability to count?

Please read it again from this point of view, and don't search for any definitions there, just try to understand this simple test.

Thank you.

Organic

cantormath

Cantor's Diagonalization Argument

Theorem-The interval [0,1] is not countably infinite.
Proof:-The proof by contradiction proceeds as follows:

Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in order). In the case of numbers with two decimal expansions, like 0.499 ... = 0.500 ..., we chose the one ending in nines. Assume, for example, that the decimal expansions of the beginning of the sequence are as follows:

r1 = 0 . 5 1 0 5 1 1 0 ...
r2 = 0 . 4 1 3 2 0 4 3 ...
r3 = 0 . 8 2 4 5 0 2 6 ...
r4 = 0 . 2 3 3 0 1 2 6 ...
r5 = 0 . 4 1 0 7 2 4 6 ...
r6 = 0 . 9 9 3 7 8 3 8 ...
r7 = 0 . 0 1 0 5 1 3 5 ...
...
We shall now construct a real number x in [0,1] by considering the kth digit after the decimal point of the decimal expansion of rk.

r1 = 0 . 5 1 0 5 1 1 0 ...
r2 = 0 . 4 1 3 2 0 4 3 ...
r3 = 0 . 8 2 4 5 0 2 6 ...
r4 = 0 . 2 3 3 0 1 2 6 ...
r5 = 0 . 4 1 0 7 2 4 6 ...
r6 = 0 . 9 9 3 7 8 3 8 ...
r7 = 0 . 0 1 0 5 1 3 5 ...
...

The digits we will consider are indicated in bold and illustrate why this is called the diagonal proof. From these digits we define the digits of x as follows.
if the kth digit of rk is 5 then the kth digit of x is 4
if the kth digit of rk is not 5 then the kth digit of x is 5
For the example above this will result in the following decimal expansion.

x = 0 . 4 5 5 5 5 5 4 ...

The number x is a real number (we know that all decimal expansions represent real numbers) in [0,1] (clearly). Hence we must have rn = x for some n, since we have assumped that ( r1, r2, r3, ... ) enumerates all real numbers in [0, 1]. However, because of the way we have chosen 4's and 5's as digits in step (6), x differs in the nth decimal place from rn, so x is not in the sequence ( r1, r2, r3, ... ). This sequence is therefore not an enumeration of the set of all reals in the interval [0,1]. This is a contradiction.

Hence the assumption that the interval [0,1] is countably infinite must be false.

Q.E.D.

It is a direct corollary of this result that the set R of all real numbers is uncountable. If R were countable, we could enumerate all of the real numbers in a sequence, and then get a sequence enumerating [0, 1] by removing all of the real numbers outside this interval. But we have just shown that this latter list cannot exist. Alternatively, we could show that [0, 1] and R are the same size by constructing a bijection between them.

Tom Mattson

Cantormath,

Welcome to Physics Forums! And you're right, there is no argument.

What you've stumbled upon here is a piece of crackpottery from the old days when we adopted an "anything goes" attitude in the Theory Development Forum. We've since tightened things up so that we only allow things that make sense.

So feel free to stick around and enjoy the Forums. Don't worry about this thread, because the orginal poster isn't even here anymore.