Hilbert's Paradox and Cantor's Diagonal Proof
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29-08-2012, 06:40 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
Starcrash: not a mathematician. Big Grin

It's not stupid, it's chicanery of utility. Wink

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29-08-2012, 07:53 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 06:38 PM)Bucky Ball Wrote:  I also respect the logic behind the count.

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29-08-2012, 08:22 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 02:33 PM)guitar_nut Wrote:  Cantor's proof shows that an infinite set of number sets cannot actually contain all number sets, because there will exist a diagonal number set not already included in the existing sets, no matter how many rows you have. Paradox.

Cantor's Diagonal Proof doesn't show, indicate, or argue, it proves that the infinity of real numbers is greater than the infinity of natural numbers.

Quote:My question is, how can you actually apply a measurement of quantity or length to something that is infinite? Doesn't a measurement, by definition, require a limitation, a starting and ending point?

Cantor was neither measuring nor counting, but comparing infinities.

Quote: Could it be that because measuring infinity creates a paradox, we actually prove that infinity DOES exist? As in, we're applying a system of finite observation (full, complete, contained) to something that is infinite, thus creating an invalid answer?

What do you mean 'exist'? Infinities exist in mathematics.

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29-08-2012, 08:28 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 02:55 PM)Bucky Ball Wrote:  I've just started to think about this junk, so I know very little. But you can do the paradox from either direction. From inside it's called the "excluded middle", I don't know if there is a term for the external paradox.

The law of the excluded middle is foundational to the definition of binary logic. It states that a proposition is either true or its negation is true. There are logics that don't include it. See W. V. O. Quine.

Quote:I have a feeling Godel's theorem could be a proof of no god, and I just started to work on it.

Godel's Theorem is lovely, a work of genius. It is about formal systems, nothing to do with God.

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29-08-2012, 08:33 PM (This post was last modified: 29-08-2012 10:55 PM by Bucky Ball.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 08:28 PM)Chas Wrote:  
(29-08-2012 02:55 PM)Bucky Ball Wrote:  I've just started to think about this junk, so I know very little. But you can do the paradox from either direction. From inside it's called the "excluded middle", I don't know if there is a term for the external paradox.

The law of the excluded middle is foundational to the definition of binary logic. It states that a proposition is either true or its negation is true. There are logics that don't include it. See W. V. O. Quine.

Quote:I have a feeling Godel's theorem could be a proof of no god, and I just started to work on it.

Godel's Theorem is lovely, a work of genius. It is about formal systems, nothing to do with God.

Thanks !
I know, but Godel was trying, towards the end, to use Ambrose's proofs, (maybe that's where he got "incompleteness" ??), I really have no clue. But I think I can use incompleteness to prove no gods.

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29-08-2012, 08:36 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 06:02 PM)Starcrash Wrote:  The topic of infinity gives me a headache. I had a very, very long discussion about whether one infinity can be larger than another. I think the very idea is stupid.

Cantor's Diagonal Method clearly proves that there are infinities of at least two sizes. There are more real numbers than rational numbers, for instance.

Quote:Hilbert's Paradox strikes me as utterly stupid. The idea of a hotel with infinite rooms ever being "full" seems to be a misunderstanding of infinity. Cantor's Paradox also seems to be a misunderstanding of infinity, as if Cantor is surprised when a new set is discovered and is added to the list... the fact that you can keep doing that forever is what makes it infinite.

In a sense, yes. Cantor's Paradox only occurs in naive set theory. Full set theory does not contain that paradox.

Quote:Anyway, here's the video that made the argument that infinities can be of different sizes. It makes the "Cantor's Paradox" argument, too.




No, the video demonstrates Cantor's diagonal method.

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29-08-2012, 10:15 PM (This post was last modified: 29-08-2012 10:18 PM by Starcrash.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 08:36 PM)Chas Wrote:  Cantor's Diagonal Method clearly proves that there are infinities of at least two sizes. There are more real numbers than rational numbers, for instance.

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).

(29-08-2012 08:36 PM)Chas Wrote:  No, the video demonstrates Cantor's diagonal method.

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

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29-08-2012, 10:17 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 06:40 PM)houseofcantor Wrote:  Starcrash: not a mathematician. Big Grin

It's not stupid, it's chicanery of utility. Wink

Actually, I am a mathematician, or at least I'm in college studying to be one. I know this subject well, and just because I disagree with the consensus doesn't make me wrong. Infinity is a difficult subject to grasp, and I think that has misled people not to think about it too hard.

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29-08-2012, 11:53 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 10:17 PM)Starcrash Wrote:  ...and just because I disagree with the consensus doesn't make me wrong.

Yeah it does. Big Grin

I kinda see it as "area" for countable infinity, and "volume" for the uncountable sort... and if the consensus makes it so we don't think about infinity too hard, it's prolly a good thing, shit will make ya nuts. Big Grin

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30-08-2012, 04:52 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 11:53 PM)houseofcantor Wrote:  
(29-08-2012 10:17 PM)Starcrash Wrote:  ...and just because I disagree with the consensus doesn't make me wrong.

Yeah it does. Big Grin

I kinda see it as "area" for countable infinity, and "volume" for the uncountable sort... and if the consensus makes it so we don't think about infinity too hard, it's prolly a good thing, shit will make ya nuts. Big Grin

Go ahead and act like a fundie all you want. If you believe I'm wrong, then please point out how... don't just dismiss it because you have an unshakable, dogmatic certainty in the conclusions of experts.

I see what you mean by area vs volume (2D vs 3D), but there aren't different dimensions involved when you're talking about strings of numbers. It's a clever way to say "there may be another bunch of numbers that you can't see from your viewpoint", but that's not the problem here. The problem lies in the incredulity surrounding the fact that there are infinite real numbers between 0 and 1.

Mathematicians readily accept the statement that "there are just as many odd numbers as there are even and odd numbers". While it would seem that this pairing trick would demonstrate that such a statement is not true, it's harder to pull this trick when both sets are orderly... infinity is intuitively easier to grasp in this situation.

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