Hilbert's Paradox and Cantor's Diagonal Proof
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04-09-2012, 04:22 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04-09-2012 04:00 PM)GirlyMan Wrote:  
(03-09-2012 08:43 AM)DLJ Wrote:  Sadly, and I hate to admit this but I don't understand the terminology and every link (within the link) that is supposed to lead to a definition of a term just takes me to more terms I don't understand.

Weeping

I haven't even worked out where the word "diagonal" means anything... very frustrating.

This is an excellent article for gaining an intuition on the "diagonalization" aspect. (pssst ... don't tell Chas I'm stealing from the NY Times again). Wink


(03-09-2012 08:43 AM)DLJ Wrote:  But I can perfectly easily follow Starcrash's explanation because that's the way I visualise it too.

Sigh!

Starcrash is just failing to appreciate that this is not a constructive proof but rather a proof by contradiction as Chas points out. The only 2 reasonable ways I can see to counter Cantor's argument is to either reject proofs by contradiction (i.e., constructivism) or reject the premise of infinity.

Other fun facts. We not only know that the cardinality of the set of real numbers is larger than the set of natural numbers, we know what it is. It's equal to the cardinality of the power set of the natural numbers (the set of all subsets of natural numbers). Which is 2 raised to the power of the size of the set of natural numbers (aka Chas' avatar aleph-naught) or aleph-one.

Well why stop there you might ask? Why indeed. What about the set of all subsets of real numbers? The cardinality of that set is 2 raised to the aleph-one power or aleph-two. And so on for well, forever.

Infinity is a mind fuck. Big Grin

Well said. Yes Good thing I don't have my reading glasses on and can't read the fine print.

I want to clarify what is meant about an RAA proof.

The construction of the list of real numbers is a complete non-issue because it is an assumption for the argument that such a list exists and is mapped.

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04-09-2012, 06:12 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(03-09-2012 01:32 PM)Vosur Wrote:  You should get your hypothesis peer-reviewed, Starcrash. I'm far from being good enough at math to have a discussion about it, but when you're so confident that your claims are true, you should publish them to a wider audience.

I'm doing exactly that. I just got impatient and wanted to rub it in, but Chas is clearly a fundamentalist on this issue. I hope everyone else follows my logic, though.

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04-09-2012, 06:39 PM (This post was last modified: 04-09-2012 06:50 PM by Starcrash.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04-09-2012 04:00 PM)GirlyMan Wrote:  This is an excellent article for gaining an intuition on the "diagonalization" aspect. (pssst ... don't tell Chas I'm stealing from the NY Times again). Wink

...

Starcrash is just failing to appreciate that this is not a constructive proof but rather a proof by contradiction as Chas points out. The only 2 reasonable ways I can see to counter Cantor's argument is to either reject proofs by contradiction (i.e., constructivism) or reject the premise of infinity.

You may notice that Cantor's list includes really long numbers so we don't have to picture how far down it goes, but the diagonal can't possibly go through every number. If we made up a list of every 4-digit number, you can only draw a diagonal through 4 of them. In a list of 5-digit numbers, only 5 of them. In a list of 100-digit numbers, only 100 of them. This clearly misses a huge set of numbers, and so when you use this rule you'll find that you've made a number that isn't in the diagonal, but still exists in the set.

So you may be tempted to say that in a list of infinitely-digited numbers the diagonal can pass through an infinite number of them, but let's consider the pattern. In any finite number, we find that the diagonal can only pass through a small percentage of them (in fact, an increasingly smaller percentage as the number rises). So why should we expect that a diagonal could pass through an infinite number of them when we get to infinity? It doesn't make sense.

Furthermore, by definition, the set doesn't include every real number if you can make another one. It doesn't mean that you've made a new number, but that one has been left off the list.

I've also mentioned several times now that a person can pretend this is being done to integers instead of real numbers, because you can literally do the same thing in reverse. Consider the example given in the link to the Times except with integers (through reversing the given example):

...5432118076
...3506768191
...5764582734
...0845365482

The integer generated is ...5796.

Quote:But we’re not done yet. The next step is to take this integer and change all its digits, replacing each of them with any other digit between 1 and 8. For example, we could change the 6 to a 3, the 9 to a 2, the 7 to a 5, and so on.

This new integer ...523 is the killer. It’s certainly not in Room 1, since it has a different first digit from the number there. It’s also not in Room 2, since its second digit disagrees. In general, it differs from the nth number in the nth place. So it doesn’t appear anywhere on the list!

You'll notice I took the example exactly and just reversed it so that we're talking about integers. Do you see now why we aren't generating new numbers? If you can't make a new integer using exactly this same standard, then you can't make a new real number unless you're using a double standard. This is why I asked Chas to give me an actual example of a new number created using this method, so I could demonstrate exactly this. Do you get now what I'm trying to say about treating only one set like infinity but the other as finite?

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04-09-2012, 06:48 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(03-09-2012 02:04 PM)Chas Wrote:  I have no idea what you are trying to do with mapping integers to integers. That is not part of the proof.

I didn't explain that well. Let me try again, but better.

You claim that .01 = .0999... and this appeara to be true. That would mean that both .01 and .0999... are mapped to 10, so we have uneven sets, right? Or not. Doing this leaves ....9990 unmapped. Every time you make a 2-to-1 mapping this way you leave an integer with a 0-to-1 mapping. The ratio of numbers still remains 1:1, and that means there are still the same number of numbers in both sets.

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04-09-2012, 09:06 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04-09-2012 06:12 PM)Starcrash Wrote:  
(03-09-2012 01:32 PM)Vosur Wrote:  You should get your hypothesis peer-reviewed, Starcrash. I'm far from being good enough at math to have a discussion about it, but when you're so confident that your claims are true, you should publish them to a wider audience.

I'm doing exactly that. I just got impatient and wanted to rub it in, but Chas is clearly a fundamentalist on this issue. I hope everyone else follows my logic, though.

No, you are tilting at a windmill. The proof assumes the mapping - there is nothing to question there.

It is a reductio ad absurdum proof. We are proving that the mapping is not true by deriving a contradiction when assuming it is true.

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04-09-2012, 09:22 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04-09-2012 06:39 PM)Starcrash Wrote:  This is why I asked Chas to give me an actual example of a new number created using this method, so I could demonstrate exactly this. Do you get now what I'm trying to say about treating only one set like infinity but the other as finite?

It's not a constructive proof. All it needs to demonstrate is an irrefutable counterexample. Only thing I can say is that what I want you to get is the point that Cantor first assumed the existence of infinity and then assumed a bijective one-to-one mapping between the natural numbers and the real numbers and then showed that there is a real number (hell, a fucking lot of real numbers, hell an uncountable number of them for that matter) which can't be mapped to them. That's all, brother. ... And if you admit the premise of infinity, you admit a motherfucking infinity of infinities. ... Not sure what else to say.

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04-09-2012, 09:39 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04-09-2012 06:39 PM)Starcrash Wrote:  
(04-09-2012 04:00 PM)GirlyMan Wrote:  This is an excellent article for gaining an intuition on the "diagonalization" aspect. (pssst ... don't tell Chas I'm stealing from the NY Times again). Wink

...

Starcrash is just failing to appreciate that this is not a constructive proof but rather a proof by contradiction as Chas points out. The only 2 reasonable ways I can see to counter Cantor's argument is to either reject proofs by contradiction (i.e., constructivism) or reject the premise of infinity.

You may notice that Cantor's list includes really long numbers so we don't have to picture how far down it goes, but the diagonal can't possibly go through every number. If we made up a list of every 4-digit number, you can only draw a diagonal through 4 of them. In a list of 5-digit numbers, only 5 of them. In a list of 100-digit numbers, only 100 of them. This clearly misses a huge set of numbers, and so when you use this rule you'll find that you've made a number that isn't in the diagonal, but still exists in the set.

So you may be tempted to say that in a list of infinitely-digited numbers the diagonal can pass through an infinite number of them, but let's consider the pattern. In any finite number, we find that the diagonal can only pass through a small percentage of them (in fact, an increasingly smaller percentage as the number rises). So why should we expect that a diagonal could pass through an infinite number of them when we get to infinity? It doesn't make sense.

Furthermore, by definition, the set doesn't include every real number if you can make another one. It doesn't mean that you've made a new number, but that one has been left off the list.

I've also mentioned several times now that a person can pretend this is being done to integers instead of real numbers, because you can literally do the same thing in reverse. Consider the example given in the link to the Times except with integers (through reversing the given example):

...5432118076
...3506768191
...5764582734
...0845365482

The integer generated is ...5796.

Quote:But we’re not done yet. The next step is to take this integer and change all its digits, replacing each of them with any other digit between 1 and 8. For example, we could change the 6 to a 3, the 9 to a 2, the 7 to a 5, and so on.

This new integer ...523 is the killer. It’s certainly not in Room 1, since it has a different first digit from the number there. It’s also not in Room 2, since its second digit disagrees. In general, it differs from the nth number in the nth place. So it doesn’t appear anywhere on the list!

You'll notice I took the example exactly and just reversed it so that we're talking about integers. Do you see now why we aren't generating new numbers? If you can't make a new integer using exactly this same standard, then you can't make a new real number unless you're using a double standard. This is why I asked Chas to give me an actual example of a new number created using this method, so I could demonstrate exactly this. Do you get now what I'm trying to say about treating only one set like infinity but the other as finite?

You are missing the whole point. Entirely.

The mapping doesn't matter, we assume it.

The point is that we assume proposition A and derive ~A, its negation. A -> ~A.

This is one of the most elegant RAA proofs in all of mathematics.

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04-09-2012, 09:51 PM (This post was last modified: 04-09-2012 09:57 PM by GirlyMan.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04-09-2012 09:39 PM)Chas Wrote:  
(04-09-2012 06:39 PM)Starcrash Wrote:  You may notice that Cantor's list includes really long numbers so we don't have to picture how far down it goes, but the diagonal can't possibly go through every number. If we made up a list of every 4-digit number, you can only draw a diagonal through 4 of them. In a list of 5-digit numbers, only 5 of them. In a list of 100-digit numbers, only 100 of them. This clearly misses a huge set of numbers, and so when you use this rule you'll find that you've made a number that isn't in the diagonal, but still exists in the set.

So you may be tempted to say that in a list of infinitely-digited numbers the diagonal can pass through an infinite number of them, but let's consider the pattern. In any finite number, we find that the diagonal can only pass through a small percentage of them (in fact, an increasingly smaller percentage as the number rises). So why should we expect that a diagonal could pass through an infinite number of them when we get to infinity? It doesn't make sense.

Furthermore, by definition, the set doesn't include every real number if you can make another one. It doesn't mean that you've made a new number, but that one has been left off the list.

I've also mentioned several times now that a person can pretend this is being done to integers instead of real numbers, because you can literally do the same thing in reverse. Consider the example given in the link to the Times except with integers (through reversing the given example):

...5432118076
...3506768191
...5764582734
...0845365482

The integer generated is ...5796.


You'll notice I took the example exactly and just reversed it so that we're talking about integers. Do you see now why we aren't generating new numbers? If you can't make a new integer using exactly this same standard, then you can't make a new real number unless you're using a double standard. This is why I asked Chas to give me an actual example of a new number created using this method, so I could demonstrate exactly this. Do you get now what I'm trying to say about treating only one set like infinity but the other as finite?

You are missing the whole point. Entirely.

The mapping doesn't matter, we assume it.

The point is that we assume proposition A and derive ~A, its negation. A -> ~A.

This is one of the most elegant RAA proofs in all of mathematics.

It's like one of the canonical examples. ... If you don't accept it you are a strict constructivist and deny proof by contradiction.

As it was in the beginning is now and ever shall be, world without end. Amen.
And I will show you something different from either
Your shadow at morning striding behind you
Or your shadow at evening rising to meet you;
I will show you fear in a handful of dust.
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