Hilbert's Paradox and Cantor's Diagonal Proof
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29-08-2012, 02:33 PM
Hilbert's Paradox and Cantor's Diagonal Proof
So I've been reading a lot of arguments for and against infinity, specifically the two in the subject. A few theists, most famously Craig, have used similar arguments to show the universe has a definite beginning. Here's a reference if you're interested:

http://en.wikipedia.org/wiki/Hilbert%27s_Hotel
http://en.wikipedia.org/wiki/Diagonal_proof

One of the philosophical arguments against infinity, if I understand these correctly, is that something infinite should be able to contain everything, but in being infinite, it creates a paradox. For example, Hilbert's hotel states that if a hotel with infinite rooms is full, it can still accommodate more guests as there will always be more rooms. Paradox. 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.

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? 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?

...it would rather be a man... [who] plunges into scientific questions with which he has no real acquaintance, only to obscure them with aimless rhetoric, and distract the attention of his hearers from the real point at issue by eloquent digressions and skilled appeals to religious prejudice.
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29-08-2012, 02:50 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
Hey, GN.

I always begin from a simple place. To me, existence makes no sense. If the universe is infinite, then what the fuck does that even mean? If the universe is finite, then what's beyond it? Neither makes sense to me.

To answer your question, I'm no mathematician so I'll skip the number stuff. I would say that yes, measurement requires a limitation; both starting and end. For example, if we count to infinity, we begin at 0 or 1. So there's a limit on one end, but infinity on the other. So, again, yes. Measurement requires limitation.

This goes back to my simple place. Counting to infinity makes no sense. And if the universe has a beginning and an end, then what is outside of it?

I don't understand the measuring infinity paradox, so I'll bow out on that one.

I can't see how something finite can measure something infinite. Above all else, energy is required to accomplish the task and if energy itself is finite, then you'll run out of gas at some point. Kind of like E=MC2. By the time you actually get to C, you need infinite energy.

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29-08-2012, 02:55 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 02:33 PM)guitar_nut Wrote:  So I've been reading a lot of arguments for and against infinity, specifically the two in the subject. A few theists, most famously Craig, have used similar arguments to show the universe has a definite beginning. Here's a reference if you're interested:

http://en.wikipedia.org/wiki/Hilbert%27s_Hotel
http://en.wikipedia.org/wiki/Diagonal_proof

One of the philosophical arguments against infinity, if I understand these correctly, is that something infinite should be able to contain everything, but in being infinite, it creates a paradox. For example, Hilbert's hotel states that if a hotel with infinite rooms is full, it can still accommodate more guests as there will always be more rooms. Paradox. 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.

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? 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?

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.

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

Craig has been debunked. He knows no math. The set he describes, formally in Math, is "undefined". He's in it for the money.

You are also assuming, (right ?), only 2 dimensions ?

"This sentence is false" .... is it true, or is it false ? ..therein lies the key. But I'm not sure how yet.

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Isaiah 45:7 "I form the light, and create darkness: I make peace, and create evil: I the LORD do all these things" (KJV)

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29-08-2012, 03:01 PM (This post was last modified: 29-08-2012 03:13 PM by houseofcantor.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
It's mathematical chicanery. Physics doesn't do "infinite."

Here's some chicanery: http://www.intmath.com//series-binomial-...series.php Can use this formula to show .999...=1.

Lemme tell you about that math stuff: it is best to compartmentalize. Have a brain part for math and a brain part for philosophy; and try not to mix the two. Thumbsup

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29-08-2012, 03:08 PM (This post was last modified: 29-08-2012 03:28 PM by cufflink.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 02:33 PM)guitar_nut Wrote:  So I've been reading a lot of arguments for and against infinity, specifically the two in the subject. A few theists, most famously Craig, have used similar arguments to show the universe has a definite beginning. Here's a reference if you're interested:

http://en.wikipedia.org/wiki/Hilbert%27s_Hotel
http://en.wikipedia.org/wiki/Diagonal_proof

One of the philosophical arguments against infinity, if I understand these correctly, is that something infinite should be able to contain everything, but in being infinite, it creates a paradox. For example, Hilbert's hotel states that if a hotel with infinite rooms is full, it can still accommodate more guests as there will always be more rooms. Paradox. 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.

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? 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 Cantor showed in his diagonal proof is that there are different orders of infinity. That may defy common sense, but there's nothing logically paradoxical about it.

Here are some basic examples. Consider these sets:

N = { 1, 2, 3, 4, . . . } That is, the set of natural numbers.

E = { 2, 4, 6, 8, . . . } That is, the set of even natural numbers.

T = { 10, 20, 30, 40, . . . } That is, the set of natural numbers that are multiples of 10.

F = { the set of all possible fractions, where numerator and denominator are natural numbers }

N, E, T, and F are clearly infinite sets. But they each contain exactly the same number of elements! (For example, N does not contain twice as many elements as E.) The reason is that the members of any one of these sets can be put into one-to-one correspondence with the members of any other.

But some infinite sets of numbers actually do contain more elements than any of the sets above. For example:

P = { all the numbers between 0 and 1 }

P includes numbers like 0.5, 0.31313131..., half the square root of 2, one quarter of pi, etc.

P is an infinite set. But it turns out that it contains more numbers than N, E, T, or F. That's what Cantor showed in his diagonal proof: the members of P cannot be put into 1-1 correspondence with the members of N. Some infinities are bigger than others, in a mathematically precise way. That's extremely interesting, even astonishing, but it's not a paradox.

I wouldn't spend too much time trying to tell if there's anything to Craig's arguments. He's not a physicist or a mathematician; he's a Christian apologist--and an unusually dishonest one at that.

Religious disputes are like arguments in a madhouse over which inmate really is Napoleon.
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29-08-2012, 03:28 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
Oh, I don't take Craig seriously. I was reading up on his side of the argument and found these proofs. My guess is he was a theist long before he got his education, and entered into his studies looking to gather information to support the conclusion he'd already come to. He makes some pretty incredible leaps and assumptions, while dazzling his audience with complex vocabulary and concepts.

He can reference quantum fields and the semantics of "nothing" all he wants... the foundation of his ideas comes from a 2000 year old mythology. Nuff said.

...it would rather be a man... [who] plunges into scientific questions with which he has no real acquaintance, only to obscure them with aimless rhetoric, and distract the attention of his hearers from the real point at issue by eloquent digressions and skilled appeals to religious prejudice.
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29-08-2012, 06:02 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
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.

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.

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




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29-08-2012, 06:07 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.

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.

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




Just to clarify, the "like" was for the video. Smartass

Religious disputes are like arguments in a madhouse over which inmate really is Napoleon.
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29-08-2012, 06:15 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 06:07 PM)cufflink Wrote:  Just to clarify, the "like" was for the video. Smartass

It has a "like" button, you know Rolleyes

I love minutephysics, and I respect the logic behind the count, but I think it takes advantage of the incomprehensibility of infinity. If all of the possible numbers between 0 and 1 are there, you don't make any "new" numbers by taking diagonal sets... the number is already there, and it just isn't noticed because not all of them have been written down to be seen. The fact that you can keep generating new numbers is just what makes it infinity by definition, and liberties are being taken with the set of 0 to 1 that isn't being taken with the set of integers (because that, too, can keep generating "new" numbers forever).

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29-08-2012, 06:38 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 06:15 PM)Starcrash Wrote:  
(29-08-2012 06:07 PM)cufflink Wrote:  Just to clarify, the "like" was for the video. Smartass

It has a "like" button, you know Rolleyes

I love minutephysics, and I respect the logic behind the count, but I think it takes advantage of the incomprehensibility of infinity. If all of the possible numbers between 0 and 1 are there, you don't make any "new" numbers by taking diagonal sets... the number is already there, and it just isn't noticed because not all of them have been written down to be seen. The fact that you can keep generating new numbers is just what makes it infinity by definition, and liberties are being taken with the set of 0 to 1 that isn't being taken with the set of integers (because that, too, can keep generating "new" numbers forever).

I also respect the logic behind the count.

http://ts3.mm.bing.net/th?id=I4847038793...8&c=7&rs=1

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Isaiah 45:7 "I form the light, and create darkness: I make peace, and create evil: I the LORD do all these things" (KJV)

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