Hilbert's Paradox and Cantor's Diagonal Proof
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02-09-2012, 06:06 PM (This post was last modified: 02-09-2012 06:14 PM by fstratzero.)
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 04:10 PM)Starcrash Wrote:  
(30-08-2012 07:21 AM)Chas Wrote:  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.

I meant to post it today, but it'll take a day... tomorrow I plan to post a nice YouTube video explaining how the mathematical consensus on infinity is plain fucking wrong.

Let me explain it very shortly. The difference between positive integers and real numbers between 0 and 1 is simply the difference between what side of the decimal point they're on. Now, one might complain that switching the decimal point is problematic because of zeroes... 0.1 and 0.01 both become "1" when you move the decimal to the other side of the number (just as 10 and 100 both become .1 when going the other way), but if you reverse the order of the numbers before moving the decimal point, then you get one and only one match from the other set.

For example, .8274 becomes 4728 and vice versa. 1000 becomes .0001 and vice versa. Each and every positive integer has one and only one match, while each real number between 0 and 1 has one and only one match, and if you can match every single number between sets then you have an equal number of numbers in each set.

The problem behind the "diagonal proof" is that you could do it with the integers and come up with "new" numbers, but instead mathematicians have made a double standard by which these numbers are assumed to be used up but the real numbers are not, but as I just demonstrated, every damn number is matched between sets... the "new" numbers are there, and it wasn't realized because they hadn't been organized in any way that one could find them in the set.

You're wrong, Chas. You're wrong, HouseofCantor. Suck it up and move on.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for "working mathematicians" to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

http://en.wikipedia.org/wiki/Kurt_G%C3%B6del

Yup he tried to disprove set theory.


Perhaps that's what you are looking for?
http://en.wikipedia.org/wiki/Gyula_K%C5%91nig

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02-09-2012, 06:18 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
Gosh I'm so lucky to know you crazy math people. I'm also pretty lucky I'm kinda drunk right now or none of this crap would make me laugh this much. Shy

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02-09-2012, 06:32 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 06:18 PM)kim Wrote:  Gosh I'm so lucky to know you crazy math people. I'm also pretty lucky I'm kinda drunk right now or none of this crap would make me laugh this much. Shy

Yeah, I used to have a math lab, but the extraneous solutions and radicals were imaginary and I didn't have the means.

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02-09-2012, 06:45 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 06:06 PM)fstratzero Wrote:  Perhaps that's what you are looking for?
http://en.wikipedia.org/wiki/Gyula_K%C5%91nig

My my. That is fucking interesting zero. ... Thanks. Thumbsup

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02-09-2012, 06:46 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 04:10 PM)Starcrash Wrote:  For example, .8274 becomes 4728 and vice versa. 1000 becomes .0001 and vice versa. Each and every positive integer has one and only one match, while each real number between 0 and 1 has one and only one match, and if you can match every single number between sets then you have an equal number of numbers in each set.

Hmm.

I can see how your process works with terminating decimals--i.e., the ones that eventually "stop" and just continue on with zeroes after that, like your 0.8274 example, which is the same as 0.827400000000000... And sure, if you confine your discussion to the terminating decimals between 0 and 1, they're perfectly countable.

But there are also repeating decimals like 0.33333... (=1/3) and 0.599599599599... (=599/999) etc. And decimals where there is no pattern: irrational algebraics like .7071067811865... (half the square root of two), irrational transcendentals like 0.7853981633974... (one quarter of pi). How does your matching process work in those cases?

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02-09-2012, 07:08 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 05:42 PM)Chas Wrote:  
(02-09-2012 05:09 PM)GirlyMan Wrote:  I'd rethink this before I did something so silly and regrettable.

I think I'd better drive up to Maine to beat some sense into him.

I hear Lobster Rolls are delicious.

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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02-09-2012, 07:34 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 07:08 PM)GirlyMan Wrote:  
(02-09-2012 05:42 PM)Chas Wrote:  I think I'd better drive up to Maine to beat some sense into him.

I hear Lobster Rolls are delicious.

Yeah, fried clams on the way up, lobster roll on the way back. Good thinking.Drooling

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02-09-2012, 07:39 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 06:46 PM)cufflink Wrote:  
(02-09-2012 04:10 PM)Starcrash Wrote:  For example, .8274 becomes 4728 and vice versa. 1000 becomes .0001 and vice versa. Each and every positive integer has one and only one match, while each real number between 0 and 1 has one and only one match, and if you can match every single number between sets then you have an equal number of numbers in each set.

Hmm.

I can see how your process works with terminating decimals--i.e., the ones that eventually "stop" and just continue on with zeroes after that, like your 0.8274 example, which is the same as 0.827400000000000... And sure, if you confine your discussion to the terminating decimals between 0 and 1, they're perfectly countable.

But there are also repeating decimals like 0.33333... (=1/3) and 0.599599599599... (=599/999) etc. And decimals where there is no pattern: irrational algebraics like .7071067811865... (half the square root of two), irrational transcendentals like 0.7853981633974... (one quarter of pi). How does your matching process work in those cases?

Repeating decimals and terminating decimals are all rational numbers, and the rational numbers can be put in correspondence with the natural numbers; that is, the set of rationals and the natural numbers have the same cardinality, Aleph Null.

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02-09-2012, 08:45 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
The infinity that philosophers and mathematicians study is not a true physical infinity, but a model of what we think infinity is. For instance, we have a mathematical model for a perfect sphere without any paradoxes, but we know that a perfect sphere could never exist. The closest anything material could get to a perfect sphere would have an outer skin made of a monolayer of Hydrogen atoms.

In my slight brushing with quantum mechanics, there seems to be a physical property to the universe that corrects for all kinds of paradoxes. Quantum wave functions and quantum states act really weird in order to avoid some of the paradoxes of limited infinities. We're surrounded by infinities and we exist just fine.

Existence relies on everything being quantum or in a sort of digital state (this is a really poor description, but the best I can do. It's been a couple decades since I've solved any probability fields.) To say that it is impossible for our universe to have always existed in some unknown state is not true. Nobody knows. We DO NOT know or understand enough about infinity to make any claim about it or about what may have given birth to the big bang. This is the arrogance that I find most distasteful when dealing the the First Cause Argument. Theists and apologists are always quick to claim how impossible and ridiculous the idea of an infinite regression is when we don't even know just how weird a physical infinity could be. We see reality correcting for local paradoxes and I would accept a hypothesis that there might be a similar mechanism that corrects for the paradoxes when dealing with an infinite state before the big bang.

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02-09-2012, 09:03 PM
RE: Hilbert's Paradox and Cantor's Diagonal Proof
(02-09-2012 07:39 PM)Chas Wrote:  
(02-09-2012 06:46 PM)cufflink Wrote:  Hmm.

I can see how your process works with terminating decimals--i.e., the ones that eventually "stop" and just continue on with zeroes after that, like your 0.8274 example, which is the same as 0.827400000000000... And sure, if you confine your discussion to the terminating decimals between 0 and 1, they're perfectly countable.

But there are also repeating decimals like 0.33333... (=1/3) and 0.599599599599... (=599/999) etc. And decimals where there is no pattern: irrational algebraics like .7071067811865... (half the square root of two), irrational transcendentals like 0.7853981633974... (one quarter of pi). How does your matching process work in those cases?

Repeating decimals and terminating decimals are all rational numbers, and the rational numbers can be put in correspondence with the natural numbers; that is, the set of rationals and the natural numbers have the same cardinality, Aleph Null.

Right. The problem with Starcrash's 1-1 pairing is that he hasn't even accounted for pairing up all the rationals in that interval, much less all the reals.

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