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InvisibleDiploidM
Cuban


Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
The Size Of Infinity
    #14388930 - 05/02/11 02:28 PM (12 years, 9 months ago)

First a math primmer. Bear with me, it's not hard and it's very interesting.

In math, the word Set codifies the notion of a "group of things". For example, we can talk about the set of all capital letters in English and we're talking about all 26 upper-case letters. We say that this set contains 26 Elements. This is a Finite set because it contains a finite number of elements, exactly 26 of them.

But there are also Infinite sets. For example, the set of all Natural numbers. Natural numbers are the ordinary positive counting numbers, like 1, 2, 3, 100, 473, and so on.

Another infinite set is the set of all Real Numbers. Examples of Real Numbers are all the values between any two Natural numbers. For example, between 2 and 3 there exists 2.5, 2.1974, 2.1, and 2.99999.

In math, a Map or Mapping is a way to connect one thing to another. For example, most of us have worked a puzzle where there's a list of cities on the left and a list of states on the right. The object of the puzzle is to draw a line from each city to the state it's in. This is a Mapping of city to state.

OK, now the beef.

It is possible to Map every element in one infinite set to every element in another infinite set. Obviously, with infinite sets you can't actually write down every number and draw a line connecting them like our puzzle above, but you CAN generalize this concept. For example, you can define a mapping of every positive Natural number P with every negative Natural number N with an equation of the form:


P = -N    for all P and N being a Natural number.


So, if P is 7, then it maps to N being -7. If P is 12, then it maps to N being -12. See how that works?

I've just demonstrated that for every Natural number, there exists a negative Natural number, and I demonstrated this without having to write down every Natural number and a line connecting them, which is impossible because there are infinitely many of them. I've shown a one-to-one correspondence (a bijection in math geek talk) between the two sets with a generalized mapping.

Now here's where things get weird.

In the late 1800's, a clever guy name Georg Cantor figured out a way to map more complex types of infinite sets to each other. The famous result is called Cantor's Diagonal Theorem. With it, it is possible to map the set of Natural numbers to the set of Real numbers. The startling result was that after the mapping is complete, there are Real numbers left over with no corresponding Natural number to map to.

In other words, even though there are infinitely many Naturals and also infinitely many Reals, there are nevertheless more Reals than Naturals because some Reals are left unpaired when the mapping is done.

Weird eh? What do you make of that?


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Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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OfflineNetDiver
Wandering Mindfuck


Registered: 08/24/09
Posts: 6,024
Loc: Everywhere and Nowhere
Last seen: 1 year, 7 months
Re: The Size Of Infinity [Re: Diploid]
    #14388971 - 05/02/11 02:36 PM (12 years, 9 months ago)

I'd be very interested in knowing how he demonstrated that, despite the fact that I probably wouldn't be able to comprehend it due to my lack of detailed mathematical knowledge.

But I think the abstract, mathematical infinities that Cantor worked with (he went completely crazy from trying to figure it out BTW) are very different from the infinity in which we live (i.e. the sum of everything that is, will be, and has been). To claim otherwise would be to confuse the map with the territory.


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InvisibleDiploidM
Cuban


Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: NetDiver]
    #14389002 - 05/02/11 02:42 PM (12 years, 9 months ago)

His proof (and several more contemporary ones) are available online, but they're too technical for this thread.


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Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: NetDiver]
    #14390174 - 05/02/11 05:54 PM (12 years, 9 months ago)

Quote:

Samurai Drifter said:
I'd be very interested in knowing how he demonstrated that, despite the fact that I probably wouldn't be able to comprehend it due to my lack of detailed mathematical knowledge.

But I think the abstract, mathematical infinities that Cantor worked with (he went completely crazy from trying to figure it out BTW) are very different from the infinity in which we live (i.e. the sum of everything that is, will be, and has been). To claim otherwise would be to confuse the map with the territory.




The infinity in which we live?  I dont get it.  The sum of everything that is, will be and has been isn't infinite.  Its bounded, that is - it is finite.


A simple geometric picture of the various sizes of infinity comes from considering a circle.  Now draw a mathematical line from the center of the circle to the radius.  It would take an infinite amount of such lines to completely cover the circumference.  Now increase the size of your circle.  Suddenly, that infinite set of lines no longer covers the circumference!  You need more lines to completely cover a larger circle.  From this simple picture its clear to see that all infinities are not created equal.

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Invisible1983
Stranger

Registered: 04/14/11
Posts: 130
Re: The Size Of Infinity [Re: DieCommie]
    #14390247 - 05/02/11 06:06 PM (12 years, 9 months ago)

Eleven is one louder.


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OfflineNetDiver
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Registered: 08/24/09
Posts: 6,024
Loc: Everywhere and Nowhere
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Re: The Size Of Infinity [Re: DieCommie]
    #14391070 - 05/02/11 08:35 PM (12 years, 9 months ago)

Quote:

DieCommie said:
The infinity in which we live?  I dont get it.  The sum of everything that is, will be and has been isn't infinite.  Its bounded, that is - it is finite.



Well, we live in time. Time seems to be unbounded (or rather, the Universe seems to be "timeless:" http://www.physorg.com/news/2011-04-scientists-spacetime-dimension.html ).

It makes no sense to think of time itself as having a beginning or an end, since that invariably begs the question of what was "before" time, and what will come "after." By definition, there can be nothing before or after time, since there would be no time in which it could exist. If nothing comes before time, and nothing comes afterwards, then time is infinite, because literally everything is within it.


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InvisibleCups
technically "here"
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Registered: 12/24/09
Posts: 1,925
Re: The Size Of Infinity [Re: Diploid]
    #14391331 - 05/02/11 09:11 PM (12 years, 9 months ago)

First off- Would you say this falls under Philosophy Sociology or Psychology? :grin:

So wikipedia says-

Natural Numbers- The set of all whole numbers greater than zero.

Real Numbers- A value that represents a quantity along a continuum.

Given those two facts, I literally cannot see why there would be numbers left over.  :shrug:

Regardless of what the proof say isn't it a bunch of BS at the end of the day since even though an unpaired number may "exist" the second you describe it you simultaneously describe it's pair?  :strokebeard:


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What's up everybody?!

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InvisibleDiploidM
Cuban


Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: Cups]
    #14391420 - 05/02/11 09:25 PM (12 years, 9 months ago)

Regardless of what the proof say isn't it a bunch of BS

The proofs are correct and unasailable. Read them and tell me where the error is.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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InvisibleCups
technically "here"
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Registered: 12/24/09
Posts: 1,925
Re: The Size Of Infinity [Re: Diploid]
    #14391457 - 05/02/11 09:32 PM (12 years, 9 months ago)

I'm sure they are technically correct Diploid...I don't doubt your knowledge of mathematics.

But follow the bouncing ball here.  Proof says there are X numbers left over.  Somehow you determine what one of these numbers is.

The moment you do this is ceases to be an unpaired number because once you know the Natural number you know it's Real mate.  Right?

So it seems to me that there aren't numbers left over...so much as unknown numbers.


--------------------
What's up everybody?!

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InvisibleDiploidM
Cuban


Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: Cups]
    #14391472 - 05/02/11 09:34 PM (12 years, 9 months ago)

So it seems to me that there aren't numbers left over

That's why you should read the proofs. I'm not a math teacher.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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InvisibleCups
technically "here"
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Registered: 12/24/09
Posts: 1,925
Re: The Size Of Infinity [Re: Diploid]
    #14391527 - 05/02/11 09:43 PM (12 years, 9 months ago)

Am I wrong to say that if you knew one of these left over numbers you would instantly know it's pair?

It's a simple Yes or No...no math instruction needed.


--------------------
What's up everybody?!

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OfflineFocusHawaii
Keeper of theMagic Garden

Registered: 12/27/02
Posts: 1,013
Loc: Canada
Last seen: 12 years, 6 months
Re: The Size Of Infinity [Re: Cups]
    #14391559 - 05/02/11 09:46 PM (12 years, 9 months ago)

Quote:

Cups said:

I literally cannot see why there would be numbers left over. 




I may be misunderstanding the mapping concept, but I'll explain why I think there would be numbers left over. The numbers in the whole number set could only be mapped to the corresponding whole numbers in the real number set i.e. 2 could be mapped to 2 in both sets. However, the infinite non-whole numbers (2.1, 2.2 etc) in the real set could not be mapped as they are excluded, by definition, from the whole number set.

Is this correct?

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InvisibleCups
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Registered: 12/24/09
Posts: 1,925
Re: The Size Of Infinity [Re: FocusHawaii]
    #14391573 - 05/02/11 09:47 PM (12 years, 9 months ago)

Yeah I thought about that but it seemed too simple to be worthy of a Diploid topic.


--------------------
What's up everybody?!

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OfflineFocusHawaii
Keeper of theMagic Garden

Registered: 12/27/02
Posts: 1,013
Loc: Canada
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Re: The Size Of Infinity [Re: Cups]
    #14391603 - 05/02/11 09:51 PM (12 years, 9 months ago)

Well, the mathematical proof that this holds might be much, much more complex but it may be intuitively understandable with the above.

The really amazing part to me is that something can be infinitely large yet somehow smaller than something also infinite in size.

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InvisibleCups
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Re: The Size Of Infinity [Re: FocusHawaii]
    #14391614 - 05/02/11 09:54 PM (12 years, 9 months ago)

Congrats on your 1000th post. :thumbup:  I think you get a hat or something. :lol:


--------------------
What's up everybody?!

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InvisibledeCypher
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Re: The Size Of Infinity [Re: Cups]
    #14391833 - 05/02/11 10:28 PM (12 years, 9 months ago)

Quote:

Cups said:
I'm sure they are technically correct Diploid...I don't doubt your knowledge of mathematics.

But follow the bouncing ball here.  Proof says there are X numbers left over.  Somehow you determine what one of these numbers is.

The moment you do this is ceases to be an unpaired number because once you know the Natural number you know it's Real mate.  Right?

So it seems to me that there aren't numbers left over...so much as unknown numbers.




The essence of the proof is to show that any pairing you have is automatically incomplete: suppose that you've found a supposed pairing between the natural numbers and the real numbers between 0 and 1.  Then Cantor's proof demonstrates that no matter what your pairing is, there will always be another real number that isn't paired with any natural number.  Even if you adjust your pairing to include this new real number, Cantor's proof allows you to find yet ANOTHER real number that is unpaired, and so on ad infinitum.


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We are all in the gutter, but some of us are looking at the stars.

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OfflineBlueCoyote
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Re: The Size Of Infinity [Re: Diploid]
    #14393005 - 05/03/11 05:46 AM (12 years, 9 months ago)

Infinities within infinities. Yah even infinities within finites. Reminds me of fractals and quantum-space.
The 'grouping' of infinities makes them less infinite. We can make out the natural numbers 1 and 2 even when there are infinite numbers inbetween them.
Like condensing poles, making the infinite finite - zooming in and zooming out :cool:


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Though lovers be lost love shall not  And death shall have no dominion
......................................................
"Our scientific power has outrun our spiritual power. We have guided missiles and misguided men."Martin Luther King, Jr.
'Acceptance is the absolute key - at that moment you gain freedom and you gain power and you gain courage'

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Offlineazay
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Re: The Size Of Infinity [Re: FocusHawaii]
    #14393037 - 05/03/11 06:07 AM (12 years, 9 months ago)

Quote:

FocusHawaii said:
Quote:

Cups said:

I literally cannot see why there would be numbers left over. 




I may be misunderstanding the mapping concept, but I'll explain why I think there would be numbers left over. The numbers in the whole number set could only be mapped to the corresponding whole numbers in the real number set i.e. 2 could be mapped to 2 in both sets. However, the infinite non-whole numbers (2.1, 2.2 etc) in the real set could not be mapped as they are excluded, by definition, from the whole number set.

Is this correct?




This is NOT correct. Because for example the amount of rational numbers (fractions) is the same as the number of whole numbers. Even though there are an infinite amount of rational numbers between consecutive whole numbers! The key is in reordering the rational numbers.


These proofs are very simple and you don't even need high school mathematics to understand them. So I suggest to look them up :laugh:.

Edited by azay (05/03/11 06:08 AM)

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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: NetDiver]
    #14393353 - 05/03/11 08:16 AM (12 years, 9 months ago)

Quote:

Samurai Drifter said:
Quote:

DieCommie said:
The infinity in which we live?  I dont get it.  The sum of everything that is, will be and has been isn't infinite.  Its bounded, that is - it is finite.



Well, we live in time. Time seems to be unbounded (or rather, the Universe seems to be "timeless:" http://www.physorg.com/news/2011-04-scientists-spacetime-dimension.html ).

It makes no sense to think of time itself as having a beginning or an end, since that invariably begs the question of what was "before" time, and what will come "after." By definition, there can be nothing before or after time, since there would be no time in which it could exist. If nothing comes before time, and nothing comes afterwards, then time is infinite, because literally everything is within it.




That article doesn't support your claim.  Time is bounded.  There is a beginning time, and a final time.  There is no 'before' the beginning of time and there is no 'after' the end of time.  It doesnt really matter if it makes sense or not, the universe isn't required to make sense.

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InvisibleDisoRDeR
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Re: The Size Of Infinity [Re: DieCommie]
    #14394198 - 05/03/11 12:18 PM (12 years, 9 months ago)

Quote:

Weird eh? What do you make of that?




I think you need to get Aleph  :tongue:

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