Hilbert's Paradox and Cantor's Diagonal Proof
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02-09-2012, 09:17 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.

Just read the fucking proof. I already ran it by the guys who posted the video that I put up and they found the proof sound. You would too if you understood my mathematical argument.

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02-09-2012, 09:23 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 06:46 PM)cufflink Wrote:  
(02-09-2012 04:10 PM)Starcrash Wrote:  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.

Hmm.

I can see how your process works with terminating decimals--i.e., the ones that eventually "stop" and just continue on with zeroes after that, like your 0.8274 example, which is the same as 0.827400000000000... And sure, if you confine your discussion to the terminating decimals between 0 and 1, they're perfectly countable.

But there are also repeating decimals like 0.33333... (=1/3) and 0.599599599599... (=599/999) etc. And decimals where there is no pattern: irrational algebraics like .7071067811865... (half the square root of two), irrational transcendentals like 0.7853981633974... (one quarter of pi). How does your matching process work in those cases?

The numbers .8274 and .8274000000000 are the same number. Just because you can write them differently doesn't mean they appear twice in the set of real numbers. If you did convert it, it would appear as 0000000004728, which is the same as saying 4728.

A repeating decimal would convert to 3333333... etc, because you can infinitely go in either direction. There's no limit to the number of digits in an integer any more than there are limits to digits in a real number. It just "feels" like integers have an end because we read from left to right, so we don't typically write any number as repeating infinitely left.

Remember to treat both sets as infinity. They can both go infinitely away from the decimal point.

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02-09-2012, 09:24 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 09:03 PM)cufflink Wrote:  
(02-09-2012 07:39 PM)Chas Wrote:  Repeating decimals and terminating decimals are all rational numbers, and the rational numbers can be put in correspondence with the natural numbers; that is, the set of rationals and the natural numbers have the same cardinality, Aleph Null.

Right. The problem with Starcrash's 1-1 pairing is that he hasn't even accounted for pairing up all the rationals in that interval, much less all the reals.

I did, above. And besides, don't make an Argument from Ignorance.

Quote:What say, you, Stardust?

...or Ad Hominem. Just because you think you're right doesn't mean you are. If you believe that I'm wrong, you have to show evidence, not simply "write me off" because you have questions. That's the same shit we get from creationists about evolution. Once you understand, you'll knock it off.

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02-09-2012, 10:06 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 09:24 PM)Starcrash Wrote:  
Quote:What say, you, Stardust?

...or Ad Hominem. Just because you think you're right doesn't mean you are. If you believe that I'm wrong, you have to show evidence, not simply "write me off" because you have questions. That's the same shit we get from creationists about evolution. Once you understand, you'll knock it off.

That's an ad hominem? Really?

You don't understand the diagonal proof. I will be happy to help you, and tried with pointing out a flaw in your explanation - that you didn't understand the nature of the rationals and irrationals in the list.

Here is another explanation, and another.

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02-09-2012, 10:21 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 09:23 PM)Starcrash Wrote:  
(02-09-2012 06:46 PM)cufflink Wrote:  Hmm.

I can see how your process works with terminating decimals--i.e., the ones that eventually "stop" and just continue on with zeroes after that, like your 0.8274 example, which is the same as 0.827400000000000... And sure, if you confine your discussion to the terminating decimals between 0 and 1, they're perfectly countable.

But there are also repeating decimals like 0.33333... (=1/3) and 0.599599599599... (=599/999) etc. And decimals where there is no pattern: irrational algebraics like .7071067811865... (half the square root of two), irrational transcendentals like 0.7853981633974... (one quarter of pi). How does your matching process work in those cases?

The numbers .8274 and .8274000000000 are the same number. Just because you can write them differently doesn't mean they appear twice in the set of real numbers. If you did convert it, it would appear as 0000000004728, which is the same as saying 4728.

A repeating decimal would convert to 3333333... etc, because you can infinitely go in either direction. There's no limit to the number of digits in an integer any more than there are limits to digits in a real number. It just "feels" like integers have an end because we read from left to right, so we don't typically write any number as repeating infinitely left.

Remember to treat both sets as infinity. They can both go infinitely away from the decimal point.

The list of real numbers is Cantor's proof are the real numbers between 0 and 1, so there is nothing to the left of the decimal point.

We can write 0.8724 as 0.8723999999999999999999999..., they are the same number.
Proof:
Set x = 0.9999... (Equation 1)
Multiply both sides by 10000,
10x = 9.9999... (Equation 2)
Subtract Equation 1 from Equation 2,
9x = 9
x = 1
Therefore, from Equation 1,
1 = 0.9999...

Skepticism is not a position; it is an approach to claims.
Science is not a subject, but a method.
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02-09-2012, 10:40 PM (This post was last modified: 03-09-2012 12:55 AM by cufflink.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 09:23 PM)Starcrash Wrote:  
(02-09-2012 06:46 PM)cufflink Wrote:  Hmm.

I can see how your process works with terminating decimals--i.e., the ones that eventually "stop" and just continue on with zeroes after that, like your 0.8274 example, which is the same as 0.827400000000000... And sure, if you confine your discussion to the terminating decimals between 0 and 1, they're perfectly countable.

But there are also repeating decimals like 0.33333... (=1/3) and 0.599599599599... (=599/999) etc. And decimals where there is no pattern: irrational algebraics like .7071067811865... (half the square root of two), irrational transcendentals like 0.7853981633974... (one quarter of pi). How does your matching process work in those cases?

The numbers .8274 and .8274000000000 are the same number. Just because you can write them differently doesn't mean they appear twice in the set of real numbers. If you did convert it, it would appear as 0000000004728, which is the same as saying 4728.

A repeating decimal would convert to 3333333... etc, because you can infinitely go in either direction. There's no limit to the number of digits in an integer any more than there are limits to digits in a real number. It just "feels" like integers have an end because we read from left to right, so we don't typically write any number as repeating infinitely left.

Remember to treat both sets as infinity. They can both go infinitely away from the decimal point.

Here's pi/10 : 0.31415926535...

What would that "convert to" by your method?

You're not grasping any of this. Chas and I have tried to show you where you're wrong. If you don't believe us, take this material to any math professor at any university and see what he or she says. At this point I've done my best and I'm through.

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03-09-2012, 08:15 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 10:40 PM)cufflink Wrote:  
(02-09-2012 09:23 PM)Starcrash Wrote:  The numbers .8274 and .8274000000000 are the same number. Just because you can write them differently doesn't mean they appear twice in the set of real numbers. If you did convert it, it would appear as 0000000004728, which is the same as saying 4728.

A repeating decimal would convert to 3333333... etc, because you can infinitely go in either direction. There's no limit to the number of digits in an integer any more than there are limits to digits in a real number. It just "feels" like integers have an end because we read from left to right, so we don't typically write any number as repeating infinitely left.

Remember to treat both sets as infinity. They can both go infinitely away from the decimal point.

Here's pi/10 : 0.31415926535...

What would that "convert to" by your method?

You're not grasping any of this. Chas and I have tried to show you where you're wrong. If you don't believe us, take this material to any math professor at any university and see what he or she says. At this point I've done my best and I'm through.

It would "convert to" a number that go infinitely to the left of the decimal point! You don't grasp the concept of infinity, do you? You can keep going as far left as you like and never hit the limit of where an integer can go. Stop treating real numbers as infinite and integers as less than infinite!

But for the sake of a full answer, the number it would convert to is ...53562951413. You notice how it looks awkward with the ellipsis trailing to the left? As I explained before, one of the major problems with understanding how infinity works with integers is that we read from left to right, so we don't really have a system for writing a number that trails off to the left infinitely, but that's the best I can do.

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03-09-2012, 08:22 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 10:21 PM)Chas Wrote:  
(02-09-2012 09:23 PM)Starcrash Wrote:  The numbers .8274 and .8274000000000 are the same number. Just because you can write them differently doesn't mean they appear twice in the set of real numbers. If you did convert it, it would appear as 0000000004728, which is the same as saying 4728.

A repeating decimal would convert to 3333333... etc, because you can infinitely go in either direction. There's no limit to the number of digits in an integer any more than there are limits to digits in a real number. It just "feels" like integers have an end because we read from left to right, so we don't typically write any number as repeating infinitely left.

Remember to treat both sets as infinity. They can both go infinitely away from the decimal point.

The list of real numbers is Cantor's proof are the real numbers between 0 and 1, so there is nothing to the left of the decimal point.

We can write 0.8724 as 0.8723999999999999999999999..., they are the same number.
Proof:
Set x = 0.9999... (Equation 1)
Multiply both sides by 10000,
10x = 9.9999... (Equation 2)
Subtract Equation 1 from Equation 2,
9x = 9
x = 1
Therefore, from Equation 1,
1 = 0.9999...

I already know this proof. I think I explained above that I'm a math student. This is old stuff. Another way of "proving" it is that .3 repeating = one third, and 3 times both sides of the equation is .9 repeating = 1.

I get what you're stating, although you only imply it. You're saying that .9 repeating maps to 2 integers rather than one. I wonder, does 9 repeating equal 1 with infinite zeroes following it? It's tricky to write such numbers because of the direction that we read, but it would be interesting if one of us had a method for figuring it out. I'll see what I can do.

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03-09-2012, 08:25 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 10:06 PM)Chas Wrote:  
(02-09-2012 09:24 PM)Starcrash Wrote:  ...or Ad Hominem. Just because you think you're right doesn't mean you are. If you believe that I'm wrong, you have to show evidence, not simply "write me off" because you have questions. That's the same shit we get from creationists about evolution. Once you understand, you'll knock it off.

That's an ad hominem? Really?

You don't understand the diagonal proof. I will be happy to help you, and tried with pointing out a flaw in your explanation - that you didn't understand the nature of the rationals and irrationals in the list.

Here is another explanation, and another.

I do understand the diagonal proof. I mentioned that you could do it with integers, too... all you have to do is remove the decimal point, and I think you'd have to go diagonally left instead of right. Why wouldn't you be able to do this? Again, when comparing these sets, there's a tendency to treat the real numbers as infinitely trailing sets but we keep treating integers as finite. They're able to go as far left as real numbers can go right.

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03-09-2012, 08:34 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(03-09-2012 08:15 AM)Starcrash Wrote:  
(02-09-2012 10:40 PM)cufflink Wrote:  Here's pi/10 : 0.31415926535...

What would that "convert to" by your method?

You're not grasping any of this. Chas and I have tried to show you where you're wrong. If you don't believe us, take this material to any math professor at any university and see what he or she says. At this point I've done my best and I'm through.

It would "convert to" a number that go infinitely to the left of the decimal point! You don't grasp the concept of infinity, do you? You can keep going as far left as you like and never hit the limit of where an integer can go. Stop treating real numbers as infinite and integers as less than infinite!

But for the sake of a full answer, the number it would convert to is ...53562951413. You notice how it looks awkward with the ellipsis trailing to the left? As I explained before, one of the major problems with understanding how infinity works with integers is that we read from left to right, so we don't really have a system for writing a number that trails off to the left infinitely, but that's the best I can do.

Your argument has nothing to do with the diagonal method. The numbers in the list are the reals between 0 and 1. There is no left.

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Science is not a subject, but a method.
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