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Diploid
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The Size Of Infinity
#14388930 - 05/02/11 02:28 PM (12 years, 8 months ago) |
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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?
-------------------- 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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NetDiver
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Re: The Size Of Infinity [Re: Diploid]
#14388971 - 05/02/11 02:36 PM (12 years, 8 months ago) |
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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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Diploid
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Re: The Size Of Infinity [Re: NetDiver]
#14389002 - 05/02/11 02:42 PM (12 years, 8 months ago) |
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His proof (and several more contemporary ones) are available online, but they're too technical for this thread.
-------------------- 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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DieCommie

Registered: 12/11/03
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Re: The Size Of Infinity [Re: NetDiver]
#14390174 - 05/02/11 05:54 PM (12 years, 8 months ago) |
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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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1983
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Re: The Size Of Infinity [Re: DieCommie]
#14390247 - 05/02/11 06:06 PM (12 years, 8 months ago) |
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NetDiver
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Re: The Size Of Infinity [Re: DieCommie]
#14391070 - 05/02/11 08:35 PM (12 years, 8 months ago) |
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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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Cups
technically "here"


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Re: The Size Of Infinity [Re: Diploid]
#14391331 - 05/02/11 09:11 PM (12 years, 8 months ago) |
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First off- Would you say this falls under Philosophy Sociology or Psychology? 
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. 
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?
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Diploid
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Re: The Size Of Infinity [Re: Cups]
#14391420 - 05/02/11 09:25 PM (12 years, 8 months ago) |
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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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Cups
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Re: The Size Of Infinity [Re: Diploid]
#14391457 - 05/02/11 09:32 PM (12 years, 8 months ago) |
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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.
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Diploid
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Re: The Size Of Infinity [Re: Cups]
#14391472 - 05/02/11 09:34 PM (12 years, 8 months ago) |
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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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Cups
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Re: The Size Of Infinity [Re: Diploid]
#14391527 - 05/02/11 09:43 PM (12 years, 8 months ago) |
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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.
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FocusHawaii
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Re: The Size Of Infinity [Re: Cups]
#14391559 - 05/02/11 09:46 PM (12 years, 8 months ago) |
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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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Cups
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Yeah I thought about that but it seemed too simple to be worthy of a Diploid topic.
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FocusHawaii
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Re: The Size Of Infinity [Re: Cups]
#14391603 - 05/02/11 09:51 PM (12 years, 8 months ago) |
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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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Cups
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Congrats on your 1000th post. I think you get a hat or something.
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deCypher



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Re: The Size Of Infinity [Re: Cups]
#14391833 - 05/02/11 10:28 PM (12 years, 8 months ago) |
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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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BlueCoyote
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Re: The Size Of Infinity [Re: Diploid]
#14393005 - 05/03/11 05:46 AM (12 years, 8 months ago) |
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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
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azay

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

Registered: 12/11/03
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Re: The Size Of Infinity [Re: NetDiver]
#14393353 - 05/03/11 08:16 AM (12 years, 8 months ago) |
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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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DisoRDeR
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Re: The Size Of Infinity [Re: DieCommie]
#14394198 - 05/03/11 12:18 PM (12 years, 8 months ago) |
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Quote:
Weird eh? What do you make of that?
I think you need to get Aleph
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NetDiver
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Re: The Size Of Infinity [Re: DieCommie]
#14395665 - 05/03/11 06:00 PM (12 years, 8 months ago) |
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Quote:
DieCommie said: 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.
That basically supports a cyclical idea of time; that it's bounded without there being anything outside of it. There is a finite amount of surface area on the Earth, but you could continue walking around and around and around it indefinitely.
All depends on how you define "infinite," I guess time might not be infinite in the mathematical sense of an unbounded set of numbers, but it's infinite in duration, in that the totality of all that ever was, has been, or will be is within time.
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DieCommie

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Re: The Size Of Infinity [Re: NetDiver]
#14395713 - 05/03/11 06:06 PM (12 years, 8 months ago) |
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How does a beginning time and an end time support cyclic time? I dont think it does.
Regardless, as you say - time need not be infinite. And even if it was, there is every reason to expect it would have a cardinality or 'size' associated with it just like the infinities Diploid describes.
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NetDiver
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Re: The Size Of Infinity [Re: DieCommie]
#14395867 - 05/03/11 06:28 PM (12 years, 8 months ago) |
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Well, cyclical time results if time is bounded (i.e. there are a finite quantity of events), without there being anything outside of it.
Imagine us moving along the inside of a circle of finite size. It's not infinite, in the sense that it has a limited diameter, but you're always at some point on the circle- there is nothing outside of it, couldn't be anything outside of it. There is no time at which events are not occurring.
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Diploid
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Re: The Size Of Infinity [Re: Cups]
#14395875 - 05/03/11 06:29 PM (12 years, 8 months ago) |
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Yeah I thought about that but it seemed too simple to be worthy of a Diploid topic.
Dood, this stuff is 100 years old and there are many webs that explain it. Plus it's taught in elementary set theory.
I'd rather you just go read it somewhere because writing concise and clear prose on technical subjects takes a lot of work and time to get it right and have it make sense to the non-math geek.
But OK, just for you, I'll give it a shot, but I'm a little stoned so I hope it makes sense. I may have to edit it in the morning. Yeah, I write a lot when I'm stoned. 
First we need another primmer:
You already know what a Natural number is (1, 4, 17, 212, and so on).
You already know what a Real number is (3.5, 75.3234, 100.0, 0.53332, and so on).
Now you need to know what a special kind of Real number called an Irrational number is. No, it's not a PS&P regular. Irrational numbers are Real numbers that never repeat or terminate. Pi is a well-known Irrational. It starts with 3.141526... and goes on forever that way.
The key thing here is to understand that Irrational numbers are a special kind of Real number, and that they have infinitely many digits. They never end.
If you think about it, it should be obvious that there are infinitely many Irrational numbers. If I had infinite time, I could sit here and type 2.45239875093659823658961928569823645713406384167... and so on forever. Then I could type out another different never-ending Irrational, and so on forever. There are infinitely many of them.
OK, now more beef:
Consider a mapping of each Natural to some random Real. It looks like this. Ignore the parentheses () for now. I'll get to them shortly:
Code:
Natural >>> Some random Real
0 >>> 0.(3)913228 1 >>> 0.2(2)19242 2 >>> 0.35(3)4238 3 >>> 0.437(1)186 4 >>> 0.5813(5)63 5 >>> 0.05267(1)9 6 >>> 0.567728(2)
... and so on forever.
Now, notice three things.
1. This list is infinitely long since there are infinitely many Naturals. I stopped at 6 but I could have gone on forever.
2. If you consider the digits in diagonal (), then the string of them itself is an Irrational Real number. You can write it like this 0.3231512... and so on to infinity because the list is infinitely long.
3. And if you add +1 to each of those digits in the diagonal (), you get the following 0.4342623. All I've done is take the Irrational from #2 above and added +1 to each numeral to get a new Irrational.
Now think about the consequences here. There are two possibilities.
A. Our new Irrational (from #3 above) can NEVER appear in our list.
or
B. If we insert our new Irrational (from #3 above) into our list, then we can construct ANOTHER irrational by the diagonal () technique that is not in the list. And if we insert THAT ONE, then we can find yet ANOTHER one by the diagonal () technique that is not in the list. And so on ad infinitum.
No matter how many of these diagonal () Irrationals we find and add to the list, there will ALWAYS be another one we can find by the same method that is not in the list.
That is to say, at the end of the list, we can still find one more that is not in the list.
A more rigorous treatment than this informal exposé will show that in fact there are infinitely more Irrationals than there are Naturals. In math-talk we say that the Cardinality of the Irrationals (Aleph One) is greater than the Cardinality of the Naturals (Aleph Zero), by Zermelo-Fraenkel Set Theory with Choice.
There are infinitely many infinitely-large sets, by the way. The Aleph numbers go on forever so infinities larger than the prior infinity also go on forever. Like a meta-infinity. An infinity of successively infinitely-larger sets than the infinitely large sets that came before. Is this shit weird or what?
A question that then comes up is, is there a Cardinality between Aleph Zero and Aleph One, like an Aleph 0.5? This is a very interesting question. Cantor conjectured that there is no set whose Cardinality is between Aleph Zero and Aleph One. It's called the Continuum Hypothesis. Unfortunately, another clever guy named Kurt Godel showed that this is a formally undecidable question under the standard foundation of modern mathematics called Zermelo-Fraenkel Set Theory (if ZF is consistent).
The theorem is called Godel Incompleteness. Look it up. It's every bit as weird and beautiful as Cantor's result. I'll write a thread about it some day when I have the energy.
Hope that made sense. I just spent half the day constructing it. If you still don't get it, don't feel bad. It takes most people a little while to puzzle it through. There are lots of online resources that can help, or if you're really interested in the unexpected wacky beauty numbers reveal, audit a community college math class.
-------------------- 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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DieCommie

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Re: The Size Of Infinity [Re: NetDiver]
#14395885 - 05/03/11 06:30 PM (12 years, 8 months ago) |
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Quote:
Samurai Drifter said: Well, cyclical time results if time is bounded (i.e. there are a finite quantity of events), without there being anything outside of it.
Imagine us moving along the inside of a circle of finite size. It's not infinite, in the sense that it has a limited diameter, but you're always at some point on the circle- there is nothing outside of it, couldn't be anything outside of it. There is no time at which events are not occurring.
Right, but having a circle doesn't have a beginning or an end. A line on the other hand, does. Each are finite, only one is cyclic.
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NetDiver
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Re: The Size Of Infinity [Re: DieCommie]
#14395914 - 05/03/11 06:33 PM (12 years, 8 months ago) |
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But what I'm saying is that time can be finite without having a beginning or an end. There can be a limited number of events that are never not occurring... which is exactly what is meant by the claim "time is finite, but there is nothing outside of it."
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Cups
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Re: The Size Of Infinity [Re: NetDiver]
#14399107 - 05/04/11 09:57 AM (12 years, 8 months ago) |
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Quote:
I'd rather you just go read it somewhere because writing concise and clear prose on technical subjects takes a lot of work and time to get it right and have it make sense to the non-math geek.
Well, you sell yourself short because I think you did a good job. It makes perfect sense to me now. Not like you need it but I've given you 5 shrooms for taking the time.
It seems to me this more of philosophical example of how math...despite all it's strengths, is still a human way of describing the universe which has failures. It's 99.99% complete maybe but still limited by the brains that invented the system.
Rationally we know that one infinite set cannot be larger or smaller than another infinite set...but for our number system to work this has to be the case.
One the one hand that is patently absurd, but on the other math is correct so often we're willing to live with the absurdity. All that's IMO of course.
Thanks for posting these math things Diploid. The one you posted a while back about number distribution has stuck with me. Can't remember what it's called off hand, but you posted building heights I think and showed how there were more 1s than 2s, 2s than 3s etc
I've spent time thinking on that. Seems like there is an answer in there somewhere...I just can't get there.
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DieCommie

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Re: The Size Of Infinity [Re: Cups]
#14399132 - 05/04/11 10:02 AM (12 years, 8 months ago) |
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Quote:
Rationally we know that one infinite set cannot be larger or smaller than another infinite set...but for our number system to work this has to be the case.
Rationally we know that? I would say that intuitively and naively we assume that...
Quote:
The one you posted a while back about number distribution has stuck with me. Can't remember what it's called off hand, but you posted building heights I think and showed how there were more 1s than 2s, 2s than 3s etc
I've spent time thinking on that. Seems like there is an answer in there somewhere...I just can't get there.
The reason numbers are distributed like that is a function of our base of number system. We choose a number base such that values are distributed like that. (I recall that diploid doesnt believe that, but it is the case )
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Cups
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Re: The Size Of Infinity [Re: DieCommie]
#14399189 - 05/04/11 10:16 AM (12 years, 8 months ago) |
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Two roads diverged in a yellow wood...
Both went on forever.
So DieCommie, which is longer. Road #1 or Road #2?
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Diploid
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Re: The Size Of Infinity [Re: DieCommie]
#14399246 - 05/04/11 10:29 AM (12 years, 8 months ago) |
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We choose a number base such that values are distributed like that. (I recall that diploid doesnt believe that, but it is the case)
It's not just me who doesn't agree, the mathematician community generally disagrees with you too.
Especially given that Benford's Law applies in ALL NUMBER BASES (except the trivial base 1 and base 2) and is scale invariant.
You can verify this for yourself if you don't believe me.
If Benford's Law didn't work in hexadecimal, I'd have to agree with you. But it does work in hexadecimal. And it works in octal too. And in base 20, and base 21, and base 75, and base [anything except 1 and 2].
From the Wiki:
An extension of Benford's law predicts the distribution of first digits in other bases besides decimal; in fact, any base b ≥ 2. The general form 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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Cups
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Posts: 1,925
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Re: The Size Of Infinity [Re: Cups]
#14399261 - 05/04/11 10:32 AM (12 years, 8 months ago) |
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Actually that lat post got me thinking...
Suppose you did have 2 endless roads. Infinite asphalt.
The roads are parallel to each other.
Now suppose that one of the roads began 50ft before the other road began.
Would the first road be longer than the second even though both are infinitely long? Is something like that what you're getting at?
Edited: Never mind. From a certain perspective this may be true but in truth the roads are the same.
-------------------- What's up everybody?!
Edited by Cups (05/04/11 10:35 AM)
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johnm214


Registered: 05/31/07
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Re: The Size Of Infinity [Re: NetDiver]
#14399325 - 05/04/11 10:47 AM (12 years, 8 months ago) |
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Quote:
Samurai Drifter said: Well, cyclical time results if time is bounded (i.e. there are a finite quantity of events), without there being anything outside of it.
Imagine us moving along the inside of a circle of finite size. It's not infinite, in the sense that it has a limited diameter, but you're always at some point on the circle- there is nothing outside of it, couldn't be anything outside of it. There is no time at which events are not occurring.
I don't get the point here. Of course if you imagine time as moving along a circular path you will have time be cyclial because the path traced along a circle is also cyclical. I don't see how this supports your assertion any more than imagining a sandwich is a shoe makes it a shoe or similar to a shoe.
In any case, the whole idea of time being a dimension to our world means that there is nothing outside of it because everything will have a time-element to it by definition. This does not have anything to do with cyclical time or linear time. If the circle is cut and made into a line of length pi times d, their is still nothing going on in the universe without a time dimension of some value mapped by the line. This is also true if we liken the line to a circle or a sin wave and imagine it to be cyclical with respect to some quantity- nothing outside or inside of the circle exists.(which seems like a silly idea to me given what we know about thermodynamics and the relationship of world events to entropy- if the magnitude of change in entropy vs time changes at some point we sure haven't observed it).
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DieCommie

Registered: 12/11/03
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Re: The Size Of Infinity [Re: Diploid]
#14399435 - 05/04/11 11:14 AM (12 years, 8 months ago) |
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.
Edited by DieCommie (11/17/16 09:50 PM)
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Diploid
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Registered: 01/09/03
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Re: The Size Of Infinity [Re: DieCommie]
#14399537 - 05/04/11 11:42 AM (12 years, 8 months ago) |
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If we used base one million, then the height of skyscrapers would not follow benfords law
But that's the definition of Benford's Law. It applies to data sets that span an order. The bigger the span, the better it fits the predicted distribution.
It WOULD apply to buildings in base one million if we measured in micrometers, no?
It applies to some datasets but not all
Of course. But again, that's the definition of Benford's Law. Obviously, it can't apply to data sets with Gausian distribution for example, like the height of people or IQ because those data sets cluster around the mean.
-------------------- 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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DieCommie

Registered: 12/11/03
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Re: The Size Of Infinity [Re: Diploid]
#14399549 - 05/04/11 11:45 AM (12 years, 8 months ago) |
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And the extent to which is applies is based on the base we choose and also, as you point out, the unit we choose. There is no mystery or wonder about it, its a simple function of our base and unit choice alone.
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Diploid
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Re: The Size Of Infinity [Re: DieCommie]
#14399810 - 05/04/11 12:43 PM (12 years, 8 months ago) |
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There is no mystery or wonder about it
I think there's a lot of mystery and wonder.
Why would a list of building heights have 999 appear FAR, FAR less often than 1,111? Why would buildings 112 units taller be built so much more often?
I don't think is as simple as you say and the fact that mathematicians have been studying it for almost 100 years and still don't quite agree on a definitive explanation bears this out.
-------------------- 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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Diploid
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Re: The Size Of Infinity [Re: Diploid]
#14399996 - 05/04/11 01:24 PM (12 years, 8 months ago) |
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Here's an interesting perspective:
--
If this sounds kind of strange, scientists Jesús Torres, Sonsoles Fernández, Antonio Gamero, and Antonio Sola from the Universidad de Cordoba also call the feature surprising. The scientists published a letter in the European Journal of Physics called “How do numbers begin? (The first digit law),” which gives a short historical review of the law. Their paper also includes useful applications and explains that no one has been able to provide an underlying reason for the consistent frequencies.
“The Benford law has been an intriguing question for me for years, ever since I read about it,” Torres, who specializes in plasma physics, told PhysOrg.com. “I have used it as a surprising example at statistical physics classes to arouse the curiosity of my pupils.”
Torres et al. explain that, before Benford, a highly esteemed astronomer named Simon Newcomb discovered the law in 1881, although Newcomb’s contemporaries did not pay much attention to his publication. Both Benford and Newcomb stumbled upon the law in the same way: while flipping through pages of a book of logarithmic tables, they noticed that the pages in the beginning of the book were dirtier than the pages at the end. This meant that their colleagues who shared the library preferred quantities beginning with the number one in their various disciplines.
Benford took this observation a step further than Newcomb, and began investigating other groups of numbers, finding that the “first digit law” emerged in groups as disparate as populations, death rates, physical and chemical constants, baseball statistics, the half-lives of radioactive isotopes, answers in a physics book, prime numbers, and Fibonacci numbers. In other words, just about any group of data obtained by using measurements satisfies the law.
On the other hand, data sets that are arbitrary and contain restrictions usually don’t follow Benford’s law. For example, lottery numbers, telephone numbers, gas prices, dates, and the weights or heights of a group of people are either random or arbitrarily assigned, and not obtained by measurement.
As Torres and his colleagues explain, scientists in the decades following Benford performed numerous studies, but discovered little more about the law other than racking up a wide variety of examples. However, scientists did discover a few curiosities. For one, when investigating second significant digits of data sets, the law still held, but with less importance. Similarly, for the third and fourth digits, the appearance of the numbers started becoming equal, leveling out at a uniform 10% for the fifth digit. A second discovery attracted even more scientific interest:
“In 1961, Pinkham discovered the first general relevant result, demonstrating that Benford’s law is scale invariant and is also the only law referring to digits which can have this scale invariance,” the scientists wrote in their letter. “That is to say, as the length of the rivers of the world in kilometers fulfill Benford’s law, it is certain that these same data expressed in miles, light years, microns or in any other length units will also fulfill it.”
Torres et al. also explain that in the last years of the 20th century, some important theoretical advances have been proven (base invariance, unicity, etc.), mainly by Ted Hill and other mathematicians. While some cases can be explained (for example, house addresses almost always start with 1’s, and lower numbers must occur before higher numbers), there is still no general justification for all examples. The scientists also explain that there is no a priori criteria that tells when a data set should or should not obey the law.
“Nowadays there are many theoretical results about the law, but some points remain in darkness,” said Torres. “Why do some numerical sets, like universal physical constants, follow the law so well? We need to know not only mathematical reasons for the law, but also characterize this set of experimental data. For example, what are their points of contact? Where they come from? Apparently, they are independent.
”I hope the general necessary and sufficient conditions will be discovered in the future—many people are interested in the law, especially economists—but I also know it could be not possible ever,” he added, mentioning Godel.
physorg.com
-------------------- 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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Diploid
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Re: The Size Of Infinity [Re: DieCommie]
#14400134 - 05/04/11 02:05 PM (12 years, 8 months ago) |
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There is no mystery or wonder about it, its a simple function of our base and unit choice alone.
BTW, if this were true, wouldn't the effect be seen in a collection of random numbers spanning several orders?
But it's not. It shows up in a list of the first thousand prime numbers, but it does not show up in a random collection of numbers spanning the same range.
-------------------- 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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ssc18
Super Saiya-jin Android


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Re: The Size Of Infinity [Re: Diploid]
#14400932 - 05/04/11 04:55 PM (12 years, 8 months ago) |
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Its a non-unit super-flow blobular representation of 0 living inside all numbers.
-------------------- :*: :*: :*:
To be what you want to become you must deny what you need to become.
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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: Diploid]
#14400936 - 05/04/11 04:56 PM (12 years, 8 months ago) |
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I dont think modern mathematicians are studying it. There isnt much to study, its what you would expect if you think about it for a bit. I dont see how it could be any other way...
Quote:
Why would a list of building heights have 999 appear FAR, FAR less often than 1,111? Why would buildings 112 units taller be built so much more often?
That isnt how it works. There arn't more buildings with a height of 1111 than 999. But, considering buildings randomly distributed between 0 and 1999 units, over half of them would start with 1 because all the building between 100 and 199 as well as 1000 and 1999 units start with a one (conveniently ignoring the leading zero of course). That is all there is to it. If you change the units or the base such that the same group of buildings now fall between 0 and 9999 you will no longer see the effect. Change the units/base such that the buildings fall between 0 and 1,999,999 and you will see the effect again.
edit - I like the first reply from your link:
Quote:
I think to say Benford's Law can't be explained by scientists is not just sensationalist, it's plain wrong. Benford's law may be unintuitive but it's been explained many times in various ways and is very well understood (but not by Torres clearly).
Edited by DieCommie (05/04/11 04:58 PM)
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ssc18
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Re: The Size Of Infinity [Re: Diploid]
#14400941 - 05/04/11 04:58 PM (12 years, 8 months ago) |
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Quote:
Diploid said: There is no mystery or wonder about it, its a simple function of our base and unit choice alone.
BTW, if this were true, wouldn't the effect be seen in a collection of random numbers spanning several orders?
But it's not. It shows up in a list of the first thousand prime numbers, but it does not show up in a random collection of numbers spanning the same range.
HAHA YEAH! Prime numbers are crazy bat shit. They cant exist! Except inside nothingsomething!
So we are talking about(numbers) OR a set of "somethings" that are not solely "somethings" but are actually "nothingsomethings"
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ssc18
Super Saiya-jin Android


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Re: The Size Of Infinity [Re: ssc18]
#14400953 - 05/04/11 04:59 PM (12 years, 8 months ago) |
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Quote:
ssc18 said:
Quote:
Diploid said: There is no mystery or wonder about it, its a simple function of our base and unit choice alone.
BTW, if this were true, wouldn't the effect be seen in a collection of random numbers spanning several orders?
But it's not. It shows up in a list of the first thousand prime numbers, but it does not show up in a random collection of numbers spanning the same range.
HAHA YEAH! Prime numbers are crazy bat shit. They cant exist! Except inside nothingsomething!
So we are talking about(numbers) OR a set of "somethings" that are not solely "somethings" but are actually "nothingsomethings"
And from this we can understand that only 1 is separate. OR IS IT?!
Muwhahaha one of the epic questions of the realm of nothingness and somethingness.
-------------------- :*: :*: :*:
To be what you want to become you must deny what you need to become.
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Diploid
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Re: The Size Of Infinity [Re: DieCommie]
#14401001 - 05/04/11 05:09 PM (12 years, 8 months ago) |
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But, considering buildings randomly distributed between 0 and 1999 units, over half of them would start with 1 because all the building between 100 and 199 as well as 1000 and 1999 units start with a one (conveniently ignoring the leading zero of course).
Of course, because you're limiting the range.
If you consider buildings randomly distributed between 0 and 9999, over 30% of them would start with 1 and only 4% would start with 9.
Your explanation doesn't explain that.
-------------------- 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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ssc18
Super Saiya-jin Android


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Re: The Size Of Infinity [Re: DieCommie]
#14401044 - 05/04/11 05:18 PM (12 years, 8 months ago) |
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Quote:
DieCommie said: I dont think modern mathematicians are studying it. There isnt much to study, its what you would expect if you think about it for a bit. I dont see how it could be any other way...
Quote:
Why would a list of building heights have 999 appear FAR, FAR less often than 1,111? Why would buildings 112 units taller be built so much more often?
That isnt how it works. There arn't more buildings with a height of 1111 than 999. But, considering buildings randomly distributed between 0 and 1999 units, over half of them would start with 1 because all the building between 100 and 199 as well as 1000 and 1999 units start with a one (conveniently ignoring the leading zero of course). That is all there is to it. If you change the units or the base such that the same group of buildings now fall between 0 and 9999 you will no longer see the effect. Change the units/base such that the buildings fall between 0 and 1,999,999 and you will see the effect again.
edit - I like the first reply from your link:
Quote:
I think to say Benford's Law can't be explained by scientists is not just sensationalist, it's plain wrong. Benford's law may be unintuitive but it's been explained many times in various ways and is very well understood (but not by Torres clearly).
Oh yeah, It is being Studied. BUt first you have to understand two--things are once on this side, and then represent no things on the other.
It is like i-m-p-o-s-s-i-b-l-e.
AND it is the drawing of infinity we might just get to watch unfold...oh wait, we already are!
SOMETIMES, i question if we are really even real. I mean made out of real stuff.,. i wonder if we are existing on/in the matrix. we 2 digits that something "real" made.
Then u wonder how that something real was made? Well it was made in the something it made. It was a portal, it made Alpha Omega! Think about it...
But also think about the possibility of a God saying, HMMM I want to make perfection, and he threw together PERFETION. He threw gother when the atom of gold was actually red in color, and he threw it together when anything u can imagine happens. ANyTHing u CAn IMagine.
Symbols? What are they but darts being thrown at an invisible wall and we are feeling/sensing that wall.
What else COULD there be?
ALl this you say? Yes I agree, but lets take a different look at nothing and something.
The something is creating, creating all things all forms of imagination and possibility. And the nothingness is taking it all in.
The nothingness is simply absorbing the information reflected by somethingness. It is God, it is truly interpreting things how they "should be."
God can be in many forms, but there is only "truely" nothingness inside of the universe.
My theory is one that I already know will be proved right later in time. The blackholes are creating all, they are the capturing mechanism of "nothingness" And here comes my theory: WHen nothingness "catches" nothingness like a blanket, it will create ways to get rid of all things in between the sheets of the blanket. Because the nothingness is what the winner wants, what the king wants, what the thing we are inside of wants, whatever u want to think:let God take u through HIS course on life, and if you aren't satisfied then you need to take HIM on YOUR course in life....Teach God, because it is possible he is waiting for his ANSWERS from U?! THINK of the possibilities guys.
But God still is saying this: Lets make all possibilities! Let's do "this" guys. Let's do "time" once. Dance.
Then BOOM there goes time! Fast? Slow? Big? Small? WHAT TYPE OF SYMBOLS CAN I CAPTURE AND TAKE BACK? WHAT SHOULD I GRAB TO TAKE WITH ME?? WHAT DOES GOD WANT TO HEAR ABOUT/what is here??
"SO symbols were there guys?"
"Yeah, God.!" , "Yeah God, It was Totally Something"
"Well can u explain it?"
"Nah, I only learned symbols when I was there... I never learned how to move as 1..."
"WHat the FUk guys? that was the first thing i was hoping u would get on ur way back, how did it all move? what was there?"
"Sorry God, I only learned how to sense the energy around me."
"Oh shit, u were handicapped"
"Yeah, God I know"
"Hey u think a bunch of gold, like as much as u can want would work out?"
"Yeah God, lets just go to sleep"
"Yeah, peace out guys"
This lesson shows you that the law of nirvana is king.
Personally I think there is more to infinity then meets the eyes.
I am talking like PLatos Forms to the MAX.
-------------------- :*: :*: :*:
To be what you want to become you must deny what you need to become.
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ssc18
Super Saiya-jin Android


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Re: The Size Of Infinity [Re: Diploid]
#14401052 - 05/04/11 05:20 PM (12 years, 8 months ago) |
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Quote:
Diploid said: But, considering buildings randomly distributed between 0 and 1999 units, over half of them would start with 1 because all the building between 100 and 199 as well as 1000 and 1999 units start with a one (conveniently ignoring the leading zero of course).
Of course, because you're limiting the range.
If you consider buildings randomly distributed between 0 and 9999, over 30% of them would start with 1 and only 4% would start with 9.
Your explanation doesn't explain that.
you don't quite view the "stuff" right here bro... "stuff" being numbers
-------------------- :*: :*: :*:
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Diploid
Cuban



Registered: 01/09/03
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Re: The Size Of Infinity [Re: ssc18]
#14401070 - 05/04/11 05:23 PM (12 years, 8 months ago) |
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ssc18, one more nonsense post and I'm banning you from PS&P for a year.
Take your incoherent, schizophrenic crap to the conspiracy forum. You've been warned about this repeatedly.
-------------------- 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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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: Diploid]
#14401117 - 05/04/11 05:37 PM (12 years, 8 months ago) |
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Quote:
Diploid said: If you consider buildings randomly distributed between 0 and 9999, over 30% of them would start with 1 and only 4% would start with 9.
I dont believe that is the case.
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Diploid
Cuban



Registered: 01/09/03
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Re: The Size Of Infinity [Re: DieCommie]
#14401119 - 05/04/11 05:39 PM (12 years, 8 months ago) |
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Wait man. That is EXACTLY what Beford's Law is.
That's why it's so freaky.
-------------------- 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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Diploid
Cuban



Registered: 01/09/03
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Re: The Size Of Infinity [Re: DieCommie]
#14401128 - 05/04/11 05:40 PM (12 years, 8 months ago) |
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Here's a repost from my other thread on this topic:
Here's an actual example. Let's look at the length (in feet) of the 20 tallest structures in the world (by category so it spans several orders):
Code:
Burj Khalifa 2,717 KVLY TV Mast 2,063 Guangzhou TV Mast 2,001 CN Tower 1,815 BREN Tower 1,516 Lualualei Transmitter Tower 1,503 Petronas Tower 1,482 Ekibstuz Power Station 1,377 Dimona Radar 1,312 Belmont Transmitter 1,272 Kiev TV Tower 1,263 Gerbrandy Tower 1,203 Yangtze River Crossing 1,137 Millau Viaduct 1,122 Nurek Dam 987 Grande Dixence Dam 935 Nanjing River Crossing 843 NTT Building 790 Hassan Mosque 751 Fuhrlander Wind Turbine 689 Niederaussem Power Station 673 St Louis Arch 656 Anaconda Smelter 630 Olympic Statium Canada 585 San Jacinto Monument 574 Chicabo Temple 570 Singapore Flyer 568 Ulm Minster 548 National Flag Square 548 Kenedy Space Center VAB 541
Now look at the first digit only. Here's that list:
2 2 2 1 1 1 1 1 1 1 1 1 1 1 9 9 8 7 7 6 6 6 6 5 5 5 5 5 5 5
Do you see 1 appearing more often than 2, and 2 appearing more often than 3 and so on until 9 which appears only twice?
Here's the distribution that always emerges:
First How Often Digit It Appears 1 30.1% 2 17.6% 3 12.5% 4 9.7% 5 7.9% 6 6.7% 7 5.8% 8 5.1% 9 4.6%
This pattern emerges if you use inches instead of feet. Or millimeters, or micrometers, or angstroms.
-------------------- 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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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: Diploid]
#14401133 - 05/04/11 05:42 PM (12 years, 8 months ago) |
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I dont think so.
Looking at the case of building height cited in the benford wikipedia, those buildings are distributed between 200 and 2800 feet so we can clearly expect those building to follow the benford pattern. Re-cast those same numbers in different units/base such that they are distributed between 0 and 9999 and you will not see the pattern. That is why its not freaky.
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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: DieCommie]
#14401161 - 05/04/11 05:52 PM (12 years, 8 months ago) |
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Ok, I just took your same list of building heights. I changed the unit to a PSP unit which equals 3.68 feet. I did this so that the building heights go up to 9999 like we talked about. I calculated how many PSPs high each of the buildings are...
For leading digits I get;
9: 1 time 8: 0 times 7: 2 times 6: 1 time 5: 4 times 4: 6 times 3: 3 times 2: 12 times 1: 1 time
Just as expected, it doesn't follow benfords law because the numbers span the entire order of magnitude. How well a set of numbers follows benfords law depends on their units and the base that it used.
edit-
And of course all of this is dependent on conveniently ignoring the leading digit of absolutely each and every measured number - zero.
Edited by DieCommie (05/04/11 06:05 PM)
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Cups
technically "here"


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Re: The Size Of Infinity [Re: DieCommie]
#14401236 - 05/04/11 06:12 PM (12 years, 8 months ago) |
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-------------------- What's up everybody?!
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Diploid
Cuban



Registered: 01/09/03
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Re: The Size Of Infinity [Re: DieCommie]
#14401309 - 05/04/11 06:28 PM (12 years, 8 months ago) |
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Small sample size.
I'm going to post a comprehensive set of examples from real world data, but it's a lot of work. I have to write an app. Counting 1's and 9's in humungus lists by hand is taking too long.
-------------------- 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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ssc18
Super Saiya-jin Android


Registered: 01/20/11
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Re: The Size Of Infinity [Re: Diploid]
#14401325 - 05/04/11 06:32 PM (12 years, 8 months ago) |
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Quote:
Diploid said: ssc18, one more nonsense post and I'm banning you from PS&P for a year.
Take your incoherent, schizophrenic crap to the conspiracy forum. You've been warned about this repeatedly.
Quote:
Diploid said: ssc18, one more nonsense post and I'm banning you from PS&P for a year.
Take your incoherent, schizophrenic crap to the conspiracy forum. You've been warned about this repeatedly.
Quote:
Diploid said: Small sample size.
I'm going to post a comprehensive set of examples from real world data, but it's a lot of work. I have to write an app. Counting 1's and 9's in humungus lists by hand is taking too long.
Diploid if you actually deleted that post back in december, I had so much into that u have no idea. did u really delete it or can I get those posts back some how! that was really not a cool move. im serious that was some dank information i had.
-------------------- :*: :*: :*:
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DieCommie

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Re: The Size Of Infinity [Re: Diploid]
#14403909 - 05/05/11 07:00 AM (12 years, 8 months ago) |
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A cute story I found that helps to explain Benford's Law,
Quote:
Dow Illustrates Benford's Law
To illustrate Benford's Law, Dr. Mark J. Nigrini offered this example: "If we think of the Dow Jones stock average as 1,000, our first digit would be 1.
"To get to a Dow Jones average with a first digit of 2, the average must increase to 2,000, and getting from 1,000 to 2,000 is a 100 percent increase.
"Let's say that the Dow goes up at a rate of about 20 percent a year. That means that it would take five years to get from 1 to 2 as a first digit.
"But suppose we start with a first digit 5. It only requires a 20 percent increase to get from 5,000 to 6,000, and that is achieved in one year.
"When the Dow reaches 9,000, it takes only an 11 percent increase and just seven months to reach the 10,000 mark, which starts with the number 1. At that point you start over with the first digit a 1, once again. Once again, you must double the number -- 10,000 -- to 20,000 before reaching 2 as the first digit.
"As you can see, the number 1 predominates at every step of the progression, as it does in logarithmic sequences."
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Diploid
Cuban



Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
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Re: The Size Of Infinity [Re: DieCommie]
#14403976 - 05/05/11 07:28 AM (12 years, 8 months ago) |
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That's an excellent explanation for increasing quantities (especially those increasing at increasing rates, like salaries), but it doesn't explain why it happens in a list of river lengths, or electric bills for a city, or physical constants, or the file sizes of all the files in a hard disk.
-------------------- 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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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: Diploid]
#14403988 - 05/05/11 07:31 AM (12 years, 8 months ago) |
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Its an explanation for quantities that grow exponentially, they are more likely to be subject to the law (It would explain your portfolio value more so than your checking account value). Otherwise, the previous explanations revolving around orders of magnitude sufficiently explain the other situations you just listed.
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Diploid
Cuban



Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
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Re: The Size Of Infinity [Re: DieCommie]
#14404100 - 05/05/11 08:11 AM (12 years, 8 months ago) |
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Otherwise, the previous explanations revolving around orders of magnitude sufficiently explain the other situations you just listed.
If that were true, it would be seen in random numbers, but it's not. Yet it's seen in measured generally random numbers like river lengths or the lengths of files on a hard drive.
I wrote a program last night to count and analyze the statistics of numerals in text file, then I ran out of energy. When I have a little more time (maybe this weekend) I'm going to try and gather a solid representative sampling and see what happens.
-------------------- 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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OrgoneConclusion
Blue Fish Group



Registered: 04/01/07
Posts: 45,414
Loc: Under the C
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Re: The Size Of Infinity [Re: Diploid]
#14404104 - 05/05/11 08:13 AM (12 years, 8 months ago) |
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Quote:
I wrote a program last night to count analyze the statistics of numerals in text file, and when I have a little more time and energy (maybe this weekend) I'm going to try and gather a solid representative sampling and see what happens.
Of you could get a girlfriend.
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Diploid
Cuban



Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
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I tried, but none of them knew how to program a computer.
-------------------- 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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OrgoneConclusion
Blue Fish Group



Registered: 04/01/07
Posts: 45,414
Loc: Under the C
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Re: The Size Of Infinity [Re: Diploid]
#14404123 - 05/05/11 08:18 AM (12 years, 8 months ago) |
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BTW, I can't believe you deleted my detailed post wherein I tied gravity into the nuclear weak-force while peaking on acid.
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Cups
technically "here"


Registered: 12/24/09
Posts: 1,925
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Re: The Size Of Infinity [Re: Diploid]
#14404163 - 05/05/11 08:30 AM (12 years, 8 months ago) |
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Quote:
Diploid said: I tried, but none of them knew how to program a computer. 
Is that geek speak for blowjob?
-------------------- What's up everybody?!
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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: Diploid]
#14405804 - 05/05/11 03:23 PM (12 years, 8 months ago) |
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Quote:
Diploid said: Otherwise, the previous explanations revolving around orders of magnitude sufficiently explain the other situations you just listed.
If that were true, it would be seen in random numbers, but it's not. Yet it's seen in measured generally random numbers like river lengths or the lengths of files on a hard drive.
It is seen in random numbers, as long as you restrict them to the proper interval. If you take random numbers between 0 and 1999 you will see it. If you take them between 0 and 9999 you will not. The same goes for measured quantities. If you measure them such that they fall on an interval like 0 to 1999 you will see it, if you measure them such that they fall on an interval between 0 and 9999 you will not.
Note that generally, we use units such that measured numbers hover around an order of magnitude. We generally take random numbers to span an entire order of magnitude.
(Also, I wonder if you know what the logical fallacy is when a data set is presented as evidence, then that same data set is used to negate the claim prompting the claimant to then dismiss the data set as too small and not indicitive... ??? )
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Diploid
Cuban



Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
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Re: The Size Of Infinity [Re: DieCommie]
#14406259 - 05/05/11 04:52 PM (12 years, 8 months ago) |
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if you measure them such that they fall on an interval between 0 and 9999 you will not
I don't think that's the case. The entire premise of the Benford Law is that that is not the case. But we'll see what my analysis turns up.
I wonder if you know what the logical fallacy is when a data set is presented as evidence, then that same data set is used to negate the claim prompting the claimant to then dismiss the data set as too small and not indicitive
It's called a Yo-Mamaism. 
Edit: I give up for now. I need several large data sets to prove my point, like the billing records for the top 10 Dow companies or something like that. I can't find anything online. The best I can find is a Wiki article about river lengths (which does conform to the Benford distribution) but it's not near enough to give a convincing analysis.
But for the record, income tax fraud has been discovered by analyzing tax returns for the Benford distribution. Fraudsters make ups random numbers in their fraudulent returns, and we already know that random numbers don't have the Benford distribution. But intuition suggests that real-world tax return values would be randomly distributed, so I think there's something to Benford.
Anyway, I wrote the analysis program for nothing, but if I ever find the right kind of data to analyze, I'll resurrect this thread and tell you NEENER NEENER! 
-------------------- 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.
Edited by Diploid (09/19/12 09:23 PM)
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OrgoneConclusion
Blue Fish Group



Registered: 04/01/07
Posts: 45,414
Loc: Under the C
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Re: The Size Of Infinity [Re: Diploid]
#14406806 - 05/05/11 06:51 PM (12 years, 8 months ago) |
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Quote:
It's called a Yo-YoMamaism.

?
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Cups
technically "here"


Registered: 12/24/09
Posts: 1,925
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Re: The Size Of Infinity [Re: Diploid]
#14417467 - 05/08/11 12:04 AM (12 years, 8 months ago) |
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Quote:
Diploid said: I don't think that's the case. The entire premise of the Benford Law is that that is not the case. But we'll see what my analysis turns up.

While we're waiting can I ask an unrelated math question?
Quote from the last scene of the final episode of Battle Star Galactica.
"Law of averages. Mathematics. Let a complex system repeat itself long enough, eventually something surprising might occur. "
Is that true?
-------------------- What's up everybody?!
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scoredon
I dont know


Registered: 04/28/09
Posts: 487
Last seen: 6 years, 2 months
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Quote:
FocusHawaii said: 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.
Gabriels Horn/Torricellis Trumpet is a solid with infinitley large surface area and a definite volume. You make it by wrapping the inverse function around the x-axis. Cool stuff. http://www.wolframalpha.com/input/?i=gabriels+horn http://en.wikipedia.org/wiki/Gabriel%27s_Horn
As for the Cantor proof,It took me a minute to understand. It makes sense in an obvious way.
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Ahimsa
µdose


Registered: 01/11/07
Posts: 1,827
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Re: The Size Of Infinity [Re: Diploid]
#14418617 - 05/08/11 08:55 AM (12 years, 8 months ago) |
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I think the reason that there can be more elements in one infinite set than in another is because any multiple of infinity equals infinity.
For example, make for every Natural number in the one set a Natural number bis, tris, etc... Like so:
N, N', N'', N''',... (example: 4, 4', 4'', 4''',...)
Now there will be an infinite number of Natural numbers too that don't map into the set of Real numbers.
The result is that both set now contain an equal infinite amount of unmappable numbers.
The reason i believe lies in the fact that any multiple of infinity equals infinity.
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Poid
Shroomery's #1 Spellir




Registered: 02/04/08
Posts: 40,372
Loc: SF Bay Area
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Re: The Size Of Infinity [Re: Ahimsa]
#14420166 - 05/08/11 03:37 PM (12 years, 8 months ago) |
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Infinity is not a real number and operations on the real numbers do not apply to infinity. There are other ways to define number systems that include "infinity", such as cardinals (see: Aleph number), in which you can do arithmetic on "infinities" but they do not, in general, have the same properties as arithmetic operations on the real numbers.
-------------------- Well I try my best to be just like I am, but everybody wants you to be just like them. -- Bob Dylan  fireworks_god said:It's one thing to simply enjoy a style of life that one enjoys, but it's another thing altogether to refer to another person's choice as "wrong" or to rationalize their behavior as being pathological or resulting from some sort of inadequacy or failing so as to create a sense of superiority or separation as yet another projection of a personal fear or control issue.
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Ahimsa
µdose


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Re: The Size Of Infinity [Re: Poid]
#14420209 - 05/08/11 03:48 PM (12 years, 8 months ago) |
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Thank you!
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dzza


Registered: 12/31/10
Posts: 143
Loc: Midwest
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Re: The Size Of Infinity [Re: Diploid]
#14421146 - 05/08/11 06:54 PM (12 years, 8 months ago) |
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Quote:
Diploid said:
Now, notice three things.
1. This list is infinitely long since there are infinitely many Naturals.
2. If you consider the digits in diagonal (), then the string of them itself is an Irrational Real number. You can write it like this 0.3231512... and so on to infinity because the list is infinitely long.
3. And if you add 1 to each of those digits in the diagonal (), you get the following 0.4342623. All I've done is take the Irrational from #2 above and added 1 to each numeral to get a new Irrational.
Now think about the consequences here. There are two possibilities.
A. Our new Irrational (from #3 above) can NEVER appear in our list.
or
B. If we insert our new Irrational (from #3 above) into our list, then we can construct ANOTHER irrational by the diagonal () technique that is not in the list. And if we insert THAT ONE, then we can find yet ANOTHER one by the diagonal () technique that is not in the list. And so on ad infinitum.
No matter how many of these diagonal () Irrationals we find and add to the list, there will ALWAYS be another one we can find by the same method that is not in the list.
That is to say, at the end of the list, we can still find one more that is not in the list.
The method in step 3 can be problematic. By adding 1 to each number you do not necessarily create a new irrational because we have that 0.99999... = 1.000 in the real numbers. A workaround would be to assign every nth digit of every nth number a value, say 3, if that digit did not already equal 3, and 4 otherwise. Then you can be assured that you avoid the paradox and create a truly distinct new number.
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LittileSkierDude
Wandering Soul



Registered: 03/15/11
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Re: The Size Of Infinity [Re: Diploid]
#14427056 - 05/09/11 09:31 PM (12 years, 8 months ago) |
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Quote:
Code:
Natural >>> Some random Real
0 >>> 0.(3)913228 1 >>> 0.2(2)19242 2 >>> 0.35(3)4238 3 >>> 0.437(1)186 4 >>> 0.5813(5)63 5 >>> 0.05267(1)9 6 >>> 0.567728(2)
ok im confused. not that i think you are wrong but i need a little help understanding. if each place value can only consist of numbers 0-9, then even if there were an infinite number of places wouldnt a number eventually repeat when using your +1 method to the numbers in parentheses? assuming an infinite number of places in one irrational is the same number of places in any other irrational of course, wouldnt it statistically HAVE to repeat eventually? i could be wrong...i still dont quite grasp what you are saying...
the best way i can grasp infinity in a real life scenario seems to be a situation in which i am holding a mirror while facing a mirror, creating an infinite number of reflections of myself...although i suppose the reflection would eventually be so small that it would be smaller than the light particles and therefore wouldnt exist....
btw this is my first post on the shroomery and i love this site
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Cups
technically "here"


Registered: 12/24/09
Posts: 1,925
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Re: The Size Of Infinity [Re: Cups]
#14467835 - 05/17/11 01:01 PM (12 years, 8 months ago) |
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KA-Bump
Quote:
Cups said:
Quote:
Diploid said: I don't think that's the case. The entire premise of the Benford Law is that that is not the case. But we'll see what my analysis turns up.

While we're waiting can I ask an unrelated math question?
Quote from the last scene of the final episode of Battle Star Galactica.
"Law of averages. Mathematics. Let a complex system repeat itself long enough, eventually something surprising might occur. "
Is that true?
-------------------- What's up everybody?!
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DieCommie

Registered: 12/11/03
Posts: 29,258
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Re: The Size Of Infinity [Re: Cups]
#14467895 - 05/17/11 01:15 PM (12 years, 8 months ago) |
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Thats not really a question that can be answered. What does 'surprising' mean? It has no mathematical definition. Note that a complex system is defined as one that, though completely deterministic, displays large variations in output as a function of small variations of input. Of course what is 'large' and 'small' is relative, so there is very much a fuzzy boundary between a chaotic and non-chaotic system.
That Battlestar Galacita quote is sci-fi techno babble, which is fun - but not meaningful.
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Cups
technically "here"


Registered: 12/24/09
Posts: 1,925
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Re: The Size Of Infinity [Re: DieCommie]
#14467912 - 05/17/11 01:19 PM (12 years, 8 months ago) |
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Werd....Thanks Diecommie
-------------------- What's up everybody?!
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