Hilbert's Paradox and Cantor's Diagonal Proof



30082012, 08:45 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:42 AM)DLJ Wrote:(30082012 07:26 AM)Chas Wrote: The Wikipedia article on Cantor's Diagonal Method is pretty good. 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 onetoone 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. 

30082012, 08:49 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 08:45 AM)houseofcantor Wrote:(30082012 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)? 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! 

30082012, 09:00 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 08:32 AM)DLJ Wrote:(30082012 08:29 AM)Chas Wrote: It's called induction. I got induction in the 8th grade. When and how did you drop out? Where has your education come from? Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

30082012, 09:03 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 08:49 AM)Bishadi Wrote:(30082012 08:45 AM)houseofcantor Wrote: Here's another one: http://www.coopertoons.com/education/dia...ument.html 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. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

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30082012, 09:07 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 09:00 AM)Chas Wrote:(30082012 08:32 AM)DLJ Wrote: Hahaha! That's just typical! 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. 

30082012, 09:10 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:45 AM)Bucky Ball Wrote: This is what I think I'm sort of thinking about, in terms of Godel's incompleteness. 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. A new type of thinking is essential if mankind is to survive and move to higher levels. ~ Albert Einstein 

30082012, 09:33 AM
(This post was last modified: 30082012 09:39 AM by cufflink.)




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:17 AM)DLJ Wrote: I'm struggling here. Maybe looking at it this way will help: For the first set abovethe nonnegative integersyou'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 nonneg. 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 setthe real numbers between 0 and 1is 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 problemthere 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. Religious disputes are like arguments in a madhouse over which inmate really is Napoleon. 

30082012, 09:42 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
@ Chas  Nice avatar.
Religious disputes are like arguments in a madhouse over which inmate really is Napoleon. 

30082012, 09:42 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 09:33 AM)cufflink Wrote: Maybe looking at it this way will help: Nice explanation, thank you. If Jesus died for our sins, why is there still sin? If man was created from dust, why is there still dust? If Americans came from Europe, why are there still Europeans? 

30082012, 09:45 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
Religious disputes are like arguments in a madhouse over which inmate really is Napoleon. 

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