Hilbert's Paradox and Cantor's Diagonal Proof



04092012, 04:22 PM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04092012 04:00 PM)GirlyMan Wrote:(03092012 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. Well said. 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 nonissue because it is an assumption for the argument that such a list exists and is mapped. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

04092012, 06:12 PM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(03092012 01:32 PM)Vosur Wrote: You should get your hypothesis peerreviewed, 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. My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her nonstick pan. 

04092012, 06:39 PM
(This post was last modified: 04092012 06:50 PM by Starcrash.)




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04092012 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). 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 4digit number, you can only draw a diagonal through 4 of them. In a list of 5digit numbers, only 5 of them. In a list of 100digit 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 infinitelydigited 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. 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? My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her nonstick pan. 

04092012, 06:48 PM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(03092012 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 2to1 mapping this way you leave an integer with a 0to1 mapping. The ratio of numbers still remains 1:1, and that means there are still the same number of numbers in both sets. My girlfriend is mad at me. Perhaps I shouldn't have tried cooking a stick in her nonstick pan. 

04092012, 09:06 PM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04092012 06:12 PM)Starcrash Wrote:(03092012 01:32 PM)Vosur Wrote: You should get your hypothesis peerreviewed, 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. 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. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

04092012, 09:22 PM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04092012 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 onetoone 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. 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. 

04092012, 09:39 PM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04092012 06:39 PM)Starcrash Wrote:(04092012 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). 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. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

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04092012, 09:51 PM
(This post was last modified: 04092012 09:57 PM by GirlyMan.)




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(04092012 09:39 PM)Chas Wrote:(04092012 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 4digit number, you can only draw a diagonal through 4 of them. In a list of 5digit numbers, only 5 of them. In a list of 100digit 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. 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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