Hilbert's Paradox and Cantor's Diagonal Proof
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30-08-2012, 07:17 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
I'm struggling here.

Why isn't bunging another number in at the dots here:
{ 0, 1, 2, 3, 4, ....}

the same as bunging another number in at the dots here:
{ 0, ... 0.1, ... 0.2, ... 0.5, ... 1} ?

(please be gentle with me, I haven't touched maths for 30 years!)

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30-08-2012, 07:21 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 10:15 PM)Starcrash Wrote:  
(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.

If we're talking math, you need to be precise.

I full well know what prove means, and Cantor's diagonal method does prove that the cardinality of the real numbers is greater than the cardinality of the natural numbers.

You belief about 'using up' the members of an infinite set is meaningless in mathematics. Induction gets us all the way there.

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30-08-2012, 07:23 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29-08-2012 08:33 PM)Bucky Ball Wrote:  
(29-08-2012 08:28 PM)Chas Wrote:  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.


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.

Have at at it, Bucky!Thumbsup I don't see how you'll get there, but I admire your willingness to take the journey.Yes

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30-08-2012, 07:26 AM
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?

dood, you had the same inquiry i did 3 decades back but i used it differently. I applied to the tangible, mathematically. I then left each variable open to time on either end "Before and after" more to define the process, versus the constants.

It may seem confusing but i am objective by nature, so i used it.

If you all want the scary realization (IMO): If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist.

There's a minf duck.

You want esoteric, i be your huckleberry!
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30-08-2012, 07:26 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:17 AM)DLJ Wrote:  I'm struggling here.

Why isn't bunging another number in at the dots here:
{ 0, 1, 2, 3, 4, ....}

the same as bunging another number in at the dots here:
{ 0, ... 0.1, ... 0.2, ... 0.5, ... 1} ?

(please be gentle with me, I haven't touched maths for 30 years!)

The Wikipedia article on Cantor's Diagonal Method is pretty good.

The basic idea is that we can use up all of the natural numbers {1, 2, 3, ...} as labels on real numbers, but when we're done, we have (an infinite number of) unlabeled real numbers left over, but we're all out of natural numbers. Clear?Blink

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30-08-2012, 07:42 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:26 AM)Chas Wrote:  
(30-08-2012 07:17 AM)DLJ Wrote:  I'm struggling here.

Why isn't bunging another number in at the dots here:
{ 0, 1, 2, 3, 4, ....}

the same as bunging another number in at the dots here:
{ 0, ... 0.1, ... 0.2, ... 0.5, ... 1} ?

(please be gentle with me, I haven't touched maths for 30 years!)



The Wikipedia article on Cantor's Diagonal Method is pretty good.

The basic idea is that we can use up all of the natural numbers {1, 2, 3, ...} as labels on real numbers, but when we're done, we have (an infinite number of) unlabeled real numbers left over, but we're all out of natural numbers. Clear?Blink

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)?

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30-08-2012, 07:45 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:26 AM)Bishadi Wrote:  
(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?

dood, you had the same inquiry i did 3 decades back but i used it differently. I applied to the tangible, mathematically. I then left each variable open to time on either end "Before and after" more to define the process, versus the constants.

It may seem confusing but i am objective by nature, so i used it.

If you all want the scary realization (IMO): If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist.

There's a minf duck.

You want esoteric, i be your huckleberry!

This is what I think I'm sort of thinking about, in terms of Godel's incompleteness.

"If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist."

There's something very wrong about that. I'm not quite sure how to express it yet.

Insufferable know-it-all.Einstein
Those who were seen dancing were thought to be insane by those who could not hear the music - Friedrich Nietzsche
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30-08-2012, 08:09 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:45 AM)Bucky Ball Wrote:  
(30-08-2012 07:26 AM)Bishadi Wrote:  "If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist."

There's something very wrong about that. I'm not quite sure how to express it yet.

It makes you a part of it!

talk about a 'spooky action at a distance'.


did ya just have a daja vu?
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30-08-2012, 08:29 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 07:42 AM)DLJ Wrote:  
(30-08-2012 07:26 AM)Chas Wrote:  The Wikipedia article on Cantor's Diagonal Method is pretty good.

The basic idea is that we can use up all of the natural numbers {1, 2, 3, ...} as labels on real numbers, but when we're done, we have (an infinite number of) unlabeled real numbers left over, but we're all out of natural numbers. Clear?Blink

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)?

It's called induction.

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Science is not a subject, but a method.
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30-08-2012, 08:32 AM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30-08-2012 08:29 AM)Chas Wrote:  
(30-08-2012 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)?

It's called induction.

Hahaha! That's just typical!

"Induction" was what we started doing when I dropped out of the education system.

I knew it would come back to haunt me.

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