Hilbert's Paradox and Cantor's Diagonal Proof
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30-08-2012, 08:45 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:42 AM)DLJ Wrote:  
(30-08-2012 07:26 AM)Chas Wrote:  The Wikipedia article on Cantor's Diagonal Method is pretty good.

The basic idea is that we can use up all of the natural numbers {1, 2, 3, ...} as labels on real numbers, but when we're done, we have (an infinite number of) unlabeled real numbers left over, but we're all out of natural numbers. Clear?Blink

Nope. I have trouble with the "use up" bit and the "when we're done" bit. How can we use up an infinite number (of natural numbers)?

Here's another one: http://www.coopertoons.com/education/dia...ument.html

Not finding a simple version that doesn't get too mathy... but they get "used up" in a one-to-one correspondence, and what happens is there's always some pesky transfinites left over. My thinking is that part of the problem is that these transfinites are ratios that snuck in from geometry - and we all know that that geometry stuff is straight witchcraft. Big Grin

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30-08-2012, 08:49 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 08:45 AM)houseofcantor Wrote:  
(30-08-2012 07:42 AM)DLJ Wrote:  Nope. I have trouble with the "use up" bit and the "when we're done" bit. How can we use up an infinite number (of natural numbers)?

Here's another one: http://www.coopertoons.com/education/dia...ument.html

Not finding a simple version that doesn't get too mathy... but they get "used up" in a one-to-one correspondence, and what happens is there's always some pesky transfinites left over. My thinking is that part of the problem is that these transfinites are ratios that snuck in from geometry - and we all know that that geometry stuff is straight witchcraft. Big Grin

couple points:

a).... funny how the 'golden ratio' is in just about everything

b) .... identifying the fractal progression, via the energy/mass is how to see your math is correct.


It's stupid easy!
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30-08-2012, 09:00 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 08:32 AM)DLJ Wrote:  
(30-08-2012 08:29 AM)Chas Wrote:  It's called induction.

Hahaha! That's just typical!

"Induction" was what we started doing when I dropped out of the education system.

I knew it would come back to haunt me.

I got induction in the 8th grade. When and how did you drop out? Where has your education come from? Consider

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Science is not a subject, but a method.
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30-08-2012, 09:03 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 08:49 AM)Bishadi Wrote:  
(30-08-2012 08:45 AM)houseofcantor Wrote:  Here's another one: http://www.coopertoons.com/education/dia...ument.html

Not finding a simple version that doesn't get too mathy... but they get "used up" in a one-to-one correspondence, and what happens is there's always some pesky transfinites left over. My thinking is that part of the problem is that these transfinites are ratios that snuck in from geometry - and we all know that that geometry stuff is straight witchcraft. Big Grin

couple points:

a).... funny how the 'golden ratio' is in just about everything

No, it's not. Sine curves, for instance.

[/quote]
b) .... identifying the fractal progression, via the energy/mass is how to see your math is correct.

[/quote]

How is that relevant?

Quote:It's stupid easy!

You have that half right. The first half.

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Science is not a subject, but a method.
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30-08-2012, 09:07 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:00 AM)Chas Wrote:  
(30-08-2012 08:32 AM)DLJ Wrote:  Hahaha! That's just typical!

"Induction" was what we started doing when I dropped out of the education system.

I knew it would come back to haunt me.

I got induction in the 8th grade. When and how did you drop out? Where has your education come from? Consider

8th grade is around 15 years old? Not sure how it works where you are.
We started to learn calculus around that age.
I dropped out a couple of years after that. Maybe it's different nowadays.

Education from life, that's all.

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30-08-2012, 09:10 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:45 AM)Bucky Ball Wrote:  This is what I think I'm sort of thinking about, in terms of Godel's incompleteness.

"If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist."

There's something very wrong about that. I'm not quite sure how to express it yet.

Think of the system as the universe, my wise compadre in armchair physics...

The universe is not closed... rather, we do not know further than can be observed of the universe, and as far as we've observed and are able to apply prior understanding to the model of the universe, we can not establish "closed". There may be "other" or "further" universe... this would be unknown at this time.

I think I'll paraphrase Chas; Go for it, baby. Wink

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30-08-2012, 09:33 AM (This post was last modified: 30-08-2012 09:39 AM by cufflink.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:17 AM)DLJ Wrote:  I'm struggling here.

Why isn't bunging another number in at the dots here:
{ 0, 1, 2, 3, 4, ....}

the same as bunging another number in at the dots here:
{ 0, ... 0.1, ... 0.2, ... 0.5, ... 1} ?

(please be gentle with me, I haven't touched maths for 30 years!)

Maybe looking at it this way will help:

For the first set above--the non-negative integers--you've implied you can list all of them through some process and be sure that every one of them is on that list. Of course it's an infinite list, but that doesn't matter. You're still sure that every non-neg. integer is somewhere on the list. And of course that's true. All you have to do is list them in order of size.

But the implication you can do that for the second set--the real numbers between 0 and 1--is false. Those internal dots give the wrong impression. If they're supposed to indicate you've listed those real numbers in size order, then what number comes right after, say, 0.2? 0.200000001? 0.20000000000001? You see the problem--there is no "next bigger" number! But Cantor's point is that no matter how you think you've come up with a list of the real numbers between 0 and 1, whether in size order or through some other method, you haven't. Cantor says, "Show me your purported listing of all the real numbers between 0 and 1, and I'll construct a real number in that interval that's not on your list!" That's the essence of his diagonalization proof.

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30-08-2012, 09:42 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
@ Chas -- Nice avatar. Cool

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30-08-2012, 09:42 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:33 AM)cufflink Wrote:  Maybe looking at it this way will help:

For the first set above--the non-negative integers--you've implied you can list all of them through some process and be sure that every one of them is on that list. Of course it's an infinite list, but that doesn't matter. You're still sure that every non-neg. integer is somewhere on the list. And of course that's true. All you have to do is list them in order of size.

But the implication you can do that for the second set--the real numbers between 0 and 1--is false. Those internal dots give the wrong impression. If they're supposed to indicate you've listed those real numbers in size order, then what number comes right after, say, 0.2? 0.200000001? 0.20000000000001? You see the problem--there is no "next bigger" number! But Cantor's point is that no matter how you think you've come up with a list of the real numbers between 0 and 1, whether in size order or through some other method, you haven't. Cantor says, "Show me your purported listing of all the real numbers between 0 and 1, and I'll construct a real number in that interval that's not on your list!" That's the essence of his diagonalization proof.

Nice explanation, thank you.

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30-08-2012, 09:45 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:42 AM)guitar_nut Wrote:  Nice explanation, thank you.

My pleasure.

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