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OfflineNetDiver
Wandering Mindfuck


Registered: 08/24/09
Posts: 6,024
Loc: Everywhere and Nowhere
Last seen: 1 year, 6 months
Re: The Size Of Infinity [Re: DieCommie]
    #14395665 - 05/03/11 06:00 PM (12 years, 8 months ago)

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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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: NetDiver]
    #14395713 - 05/03/11 06:06 PM (12 years, 8 months ago)

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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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]
    #14395867 - 05/03/11 06:28 PM (12 years, 8 months ago)

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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InvisibleDiploidM
Cuban


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Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: Cups]
    #14395875 - 05/03/11 06:29 PM (12 years, 8 months ago)

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. :grin:

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. :smirk: 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. :mushroom2:


--------------------
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]
    #14395885 - 05/03/11 06:30 PM (12 years, 8 months ago)

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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OfflineNetDiver
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Registered: 08/24/09
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Re: The Size Of Infinity [Re: DieCommie]
    #14395914 - 05/03/11 06:33 PM (12 years, 8 months ago)

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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InvisibleCups
technically "here"
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Registered: 12/24/09
Posts: 1,925
Re: The Size Of Infinity [Re: NetDiver]
    #14399107 - 05/04/11 09:57 AM (12 years, 8 months ago)

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. :smile:


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


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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: Cups]
    #14399132 - 05/04/11 10:02 AM (12 years, 8 months ago)

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 :wink:)



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InvisibleCups
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Re: The Size Of Infinity [Re: DieCommie]
    #14399189 - 05/04/11 10:16 AM (12 years, 8 months ago)

Two roads diverged in a yellow wood...

Both went on forever.

So DieCommie, which is longer.  Road #1 or Road #2?


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


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InvisibleDiploidM
Cuban


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Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: DieCommie]
    #14399246 - 05/04/11 10:29 AM (12 years, 8 months ago)

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:

Formula: 0


--------------------
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: Cups]
    #14399261 - 05/04/11 10:32 AM (12 years, 8 months ago)

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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Invisiblejohnm214
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Posts: 17,582
Loc: Americas
Re: The Size Of Infinity [Re: NetDiver]
    #14399325 - 05/04/11 10:47 AM (12 years, 8 months ago)

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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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: Diploid]
    #14399435 - 05/04/11 11:14 AM (12 years, 8 months ago)

.


Edited by DieCommie (11/17/16 09:50 PM)


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InvisibleDiploidM
Cuban


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Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: DieCommie]
    #14399537 - 05/04/11 11:42 AM (12 years, 8 months ago)

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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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: Diploid]
    #14399549 - 05/04/11 11:45 AM (12 years, 8 months ago)

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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InvisibleDiploidM
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Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: DieCommie]
    #14399810 - 05/04/11 12:43 PM (12 years, 8 months ago)

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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InvisibleDiploidM
Cuban


Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: Diploid]
    #14399996 - 05/04/11 01:24 PM (12 years, 8 months ago)

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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InvisibleDiploidM
Cuban


Folding@home Statistics
Registered: 01/09/03
Posts: 19,274
Loc: Rabbit Hole
Re: The Size Of Infinity [Re: DieCommie]
    #14400134 - 05/04/11 02:05 PM (12 years, 8 months ago)

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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Offlinessc18
Super Saiya-jin Android
Male

Registered: 01/20/11
Posts: 254
Last seen: 11 years, 10 months
Re: The Size Of Infinity [Re: Diploid]
    #14400932 - 05/04/11 04:55 PM (12 years, 8 months ago)

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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InvisibleDieCommie

Registered: 12/11/03
Posts: 29,258
Re: The Size Of Infinity [Re: Diploid]
    #14400936 - 05/04/11 04:56 PM (12 years, 8 months ago)

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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