Hilbert's Paradox and Cantor's Diagonal Proof
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30-08-2012, 09:47 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:33 AM)cufflink Wrote:  
(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.

Thank you. That was a very clear and concise explanation.

But I would retort to Mr Cantor that there is an infinite number of numbers on my list so for every one he adds to his list, I already have one to match it on my list. He is implying that my list is finite.

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30-08-2012, 10:18 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:47 AM)DLJ Wrote:  But I would retort to Mr Cantor that there is an infinite number of numbers on my list so for every one he adds to his list, I already have one to match it on my list. He is implying that my list is finite.

Well, I shouldn't try to speak for ol' Georg, but I suspect he would say something like, "I am not implying your list is finite, young man, I am saying you do not have a list at all. If you do, please show it to me. Then we can talk."

As the creator of set theory, Cantor was a hugely important mathematician. Sad life, though. Suffered from chronic depression, for which he was hospitalized several times. Bitter disputes with rival mathematicians. Lived his last years in poverty, died in a sanatorium. The Wikipedia article on him, which looks very good, mentions what some people saw as the theological implications of Cantor's work.

Religious disputes are like arguments in a madhouse over which inmate really is Napoleon.
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30-08-2012, 10:28 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:33 AM)cufflink Wrote:  
(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.

To clarify: it doesn't matter what order the list of real numbers is.

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30-08-2012, 10:29 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:42 AM)cufflink Wrote:  @ Chas -- Nice avatar. Cool

Thanks - it's in honor of this thread.

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30-08-2012, 10:34 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 10:18 AM)cufflink Wrote:  ... I suspect he would say something like, "I am not implying your list is finite, young man, I am saying you do not have a list at all. If you do, please show it to me. Then we can talk."

And I would reply "I would show you mine but I can't remember where I put it. I think it put it next to yours. Find yours and you shall find mine".

(30-08-2012 10:18 AM)cufflink Wrote:  {Sad life, chronic depression, lived in poverty}
Now, that is a set that I can find a correlation with.

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02-09-2012, 04:10 PM (This post was last modified: 02-09-2012 04:14 PM by Starcrash.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:21 AM)Chas Wrote:  
(29-08-2012 10:15 PM)Starcrash Wrote:  Yes, this is what the mathematicians think, too. But I've pointed out the flaw in their thought process -- they treat infinity as infinite sometimes, and sometimes as less than infinite. To "use up a set" of infinite numbers is not possible. The video shows them drawing lines between the sets, which draws the assumption that all of the numbers in the integer set are "used up"... however, wherever their pairing ends, that's where new numbers are drawn from. They keep making numbers in the real numbers set, but the fact that you can keep doing this is only surprising because we feel that these numbers should come to an end for some reason. Just because you can draw a diagonal through the sets to come up with a number that wasn't on your list doesn't mean a new number outside of your paired sets has been created, because you don't know what numbers are already paired. It's not a neat orderly list like the integers are. And whenever you think you've come up with a "new" number, it can be paired with an integer because there's always another integer. It's true by definition that you can always create a new number from infinity.

Be careful, too, about using the word "proves". Proof has a different meaning in the area of math, and this isn't a theory with a proof (yet, at least).


That's what I meant to say. Please don't be pedantic.

If we're talking math, you need to be precise.

I full well know what prove means, and Cantor's diagonal method does prove that the cardinality of the real numbers is greater than the cardinality of the natural numbers.

You belief about 'using up' the members of an infinite set is meaningless in mathematics. Induction gets us all the way there.

I meant to post it today, but it'll take a day... tomorrow I plan to post a nice YouTube video explaining how the mathematical consensus on infinity is plain fucking wrong.

Let me explain it very shortly. The difference between positive integers and real numbers between 0 and 1 is simply the difference between what side of the decimal point they're on. Now, one might complain that switching the decimal point is problematic because of zeroes... 0.1 and 0.01 both become "1" when you move the decimal to the other side of the number (just as 10 and 100 both become .1 when going the other way), but if you reverse the order of the numbers before moving the decimal point, then you get one and only one match from the other set.

For example, .8274 becomes 4728 and vice versa. 1000 becomes .0001 and vice versa. Each and every positive integer has one and only one match, while each real number between 0 and 1 has one and only one match, and if you can match every single number between sets then you have an equal number of numbers in each set.

The problem behind the "diagonal proof" is that you could do it with the integers and come up with "new" numbers, but instead mathematicians have made a double standard by which these numbers are assumed to be used up but the real numbers are not, but as I just demonstrated, every damn number is matched between sets... the "new" numbers are there, and it wasn't realized because they hadn't been organized in any way that one could find them in the set.

You're wrong, Chas. You're wrong, HouseofCantor. Suck it up and move on.

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02-09-2012, 04:14 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 09:47 AM)DLJ Wrote:  Thank you. That was a very clear and concise explanation.

But I would retort to Mr Cantor that there is an infinite number of numbers on my list so for every one he adds to his list, I already have one to match it on my list. He is implying that my list is finite.

Nice catch. You've noticed what I noticed when I first heard this argument -- that both sets are treated differently. Read my explanation above of why you are right.

My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her non-stick pan.
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02-09-2012, 04:25 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 04:10 PM)Starcrash Wrote:  
(30-08-2012 07:21 AM)Chas Wrote:  If we're talking math, you need to be precise.

I full well know what prove means, and Cantor's diagonal method does prove that the cardinality of the real numbers is greater than the cardinality of the natural numbers.

You belief about 'using up' the members of an infinite set is meaningless in mathematics. Induction gets us all the way there.

I meant to post it today, but it'll take a day... tomorrow I plan to post a nice YouTube video explaining how the mathematical consensus on infinity is plain fucking wrong.

Let me explain it very shortly. The difference between positive integers and real numbers between 0 and 1 is simply the difference between what side of the decimal point they're on. Now, one might complain that switching the decimal point is problematic because of zeroes... 0.1 and 0.01 both become "1" when you move the decimal to the other side of the number (just as 10 and 100 both become .1 when going the other way), but if you reverse the order of the numbers before moving the decimal point, then you get one and only one match from the other set.

For example, .8274 becomes 4728 and vice versa. 1000 becomes .0001 and vice versa. Each and every positive integer has one and only one match, while each real number between 0 and 1 has one and only one match, and if you can match every single number between sets then you have an equal number of numbers in each set.

The problem behind the "diagonal proof" is that you could do it with the integers and come up with "new" numbers, but instead mathematicians have made a double standard by which these numbers are assumed to be used up but the real numbers are not, but as I just demonstrated, every damn number is matched between sets... the "new" numbers are there, and it wasn't realized because they hadn't been organized in any way that one could find them in the set.

You're wrong, Chas. You're wrong, HouseofCantor. Suck it up and move on.

No, you misunderstand it.

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02-09-2012, 05:09 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 04:10 PM)Starcrash Wrote:  I meant to post it today, but it'll take a day... tomorrow I plan to post a nice YouTube video explaining how the mathematical consensus on infinity is plain fucking wrong.

I'd rethink this before I did something so silly and regrettable.

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02-09-2012, 05:42 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 05:09 PM)GirlyMan Wrote:  
(02-09-2012 04:10 PM)Starcrash Wrote:  I meant to post it today, but it'll take a day... tomorrow I plan to post a nice YouTube video explaining how the mathematical consensus on infinity is plain fucking wrong.

I'd rethink this before I did something so silly and regrettable.

I think I'd better drive up to Maine to beat some sense into him.
But the traffic will really suck on Labor Day weekend.
But I could stop for fried clams on the way, and steamers on the way back.
Or I could just go play golf and let him make an ass of himself.

What say, you, Stardust?

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