Infinite Universe and random number generators.
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DragonsRightWing wrote:
a "smaller infinity"??? Now my sides hurt!!!
Whats infinity take one, infinity, or a different number? ;P
I know you believe you understood what you think I said, but I am not sure you realize what you heard is not what I meant.
If infinity = infinity-1, then 0=-1, a contradiction (subtract infinity on both sides). But it gets worse: Consider the sequence { (2*x)/x }, as x diverges to infinity. It would seem that infinity / infinity = 2. Or maybe not. What if I replace my sequence by { (3*x)/x }, or for that matter { (n/x)/(1/x) } then 0/0 = n = anything you like. Infinity might have some standard notion of divergence of a sequence, but there is a reason why we don't define operator arithmetic on infinity, and don't include "infinity" on the line of real numbers, and much of it has to do with problems like this. In any case, a lot of these concepts becomes easier for anyone who has had a bit of introductory calculus. It has taken very smart people a long time to come up with suitable notions of infinity that is mathematically viable. If you look at the kind of definitions that we take for granted (e.g. the epsilon-delta definition for limit: "For every epsilon, there exists a delta such that | x - a | < delta implies | f(x) - L | < epsilon."), you can see for yourself that it's not easy to come up with an understanding of infinity that you can mathematically manipulate. (Exercise: Work out how infinity comes into play in the above definition). It gets even worse, because even though the rational numbers have infinitely many numbers between 0 and 1, it's still not nearly big enough to describe sqrt(2)/2. So take a little calculus. Some of these concepts aren't as hard as you might think. To answer your question, infinity and infinity minus one are both infinite, but they are not equal, because infinity is not equal or even unequal to anything, including infinity. It's simply undefined, and for good reason. Anyways, these are just my 2c. There are a lot of books and people out there who can describe this very much better than I can in my feeble attempt.
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You have forgotten the Physics Shop at the Other End of the Universe. There you can purchase bottled portions of infinity, along with useful frictionless surfaces and massless beams. Bottles of space, under the brand "Really really really BIG", are also available. Unfortunately, they are out of stock of bug-free code.
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If I remember correctly, the conclusion of the program was that there are an infinite number of universes of infinite size. This begs a couple of questions: where does one universe meet another (for there to be more than one universe there surely has to be a boundary and if there is a boundary they can't be infinite) and secondly, if space is infinite (to fit these infinite universes in) is the volume of space larger than the sum of its universes (so is one infinity bigger than another)? I'm going to consult my good friend Jack Daniels and have a think about this. :laugh:
No trees were harmed in the posting of this missive; however, a large number of quantum states were changed.
Digital Thunder wrote:
where does one universe meet another (for there to be more than one universe there surely has to be a boundary and if there is a boundary they can't be infinite)
Oh sheesh. Consider two pieces of paper, each of infinite area. Now stack them, one on top of the other. Sure, they have a boundary - but are they any less infinite because of this fact? No, rather their boundary is perpendicular to their infinite axes, and so there is no issue. But why do they have to even have a boundary? Lift the top one of the other, and now there is no boundary: they are parallel, and there is a gap between them. Or is the gap the boundary? Okay then, lets remove one more dimension. Let's take two lines of infinite length. We could have them be parallel, so that they never meet; or skew lines, so that even the gap between them gets difficult to define. How about perpendicular skew lines? If one runs along the x-axis and the other along the y-axis, but they are separated along z-axis, then only one point is even really classifiable as close (not totally, but things are getting farther away from having a boundary all the time). Or we can add a dimension. Let's say infinite line A flashes in and out of existence, and the next day, infinite (skew) line B flashes in and out of existence. Even more removal from having a boundary, and still infinite. So what was your point exactly?
Digital Thunder wrote:
if space is infinite (to fit these infinite universes in) is the volume of space larger than the sum of its universes (so is one infinity bigger than another)?
Eh - I think someone else already covered this reasonably well. Yes, there are infinities that are larger than others - but this isn't one of them. It's exactly the same order of infinity.
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Lloyd Atkinson☺ wrote:
if the Universe really is infinite
It isn't, more than likely.
Lloyd Atkinson☺ wrote:
there must be a random number generator that keeps producing the same number over and over again
If you have an infinite number of [fair] generators, this is possible. The problem you are having is because of the way humans perceive probability. Think of a coin tossed three times. Three heads is just as likely as three tails, two heads with one tails and two tails with one heads. All possible outcome sets have equal probability, it is just psychologically more notable if the three heads come up. This principle extends to your number generator.
Lloyd Atkinson☺ wrote:
because in an infinite Universe, however small the chance/probability is, a random number generator that produces the same number every time must exist.
Only if an infinite number if generators exist. The probability of a generator being created over a unit of space (and/or time) can change in such a way that even in an infinite universe an non-infinite number of generators exits. [edit] Wasn't this a Dilbert or XKCD cartoon? [Edit 2] Both! http://xkcd.com/221/[^] http://www.random.org/analysis/dilbert.jpg[^]
Dalek Dave: There are many words that some find offensive, Homosexuality, Alcoholism, Religion, Visual Basic, Manchester United, Butter.
modified on Saturday, February 27, 2010 1:07 PM
Keith Barrow wrote:
Think of a coin tossed three times. Three heads is just as likely as three tails, two heads with one tails and two tails with one heads.
Eh - no. Text graphics time:
3 Heads | 2 Heads | 1 Head | 0 Heads 0 Tails | 1 Tail | 2 Tails | 3 Tails ---------------------------------------------------- HHH | HHT | HTT | TTT | HTH | THT | | THH | TTH |As you can see, there are eight ways to flip a coin three times. Only one of these comes up all Heads, just as only one of them comes up all Tails. There are, however, three possibilities for each path that gives 2 of 1 and 1 of the other; therefore 2 heads with 1 tails and 1 heads with 2 tails are EACH three times more likely than all Heads or all Tails.
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If infinity = infinity-1, then 0=-1, a contradiction (subtract infinity on both sides). But it gets worse: Consider the sequence { (2*x)/x }, as x diverges to infinity. It would seem that infinity / infinity = 2. Or maybe not. What if I replace my sequence by { (3*x)/x }, or for that matter { (n/x)/(1/x) } then 0/0 = n = anything you like. Infinity might have some standard notion of divergence of a sequence, but there is a reason why we don't define operator arithmetic on infinity, and don't include "infinity" on the line of real numbers, and much of it has to do with problems like this. In any case, a lot of these concepts becomes easier for anyone who has had a bit of introductory calculus. It has taken very smart people a long time to come up with suitable notions of infinity that is mathematically viable. If you look at the kind of definitions that we take for granted (e.g. the epsilon-delta definition for limit: "For every epsilon, there exists a delta such that | x - a | < delta implies | f(x) - L | < epsilon."), you can see for yourself that it's not easy to come up with an understanding of infinity that you can mathematically manipulate. (Exercise: Work out how infinity comes into play in the above definition). It gets even worse, because even though the rational numbers have infinitely many numbers between 0 and 1, it's still not nearly big enough to describe sqrt(2)/2. So take a little calculus. Some of these concepts aren't as hard as you might think. To answer your question, infinity and infinity minus one are both infinite, but they are not equal, because infinity is not equal or even unequal to anything, including infinity. It's simply undefined, and for good reason. Anyways, these are just my 2c. There are a lot of books and people out there who can describe this very much better than I can in my feeble attempt.
I think you summed it up pretty nicely there :)
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I got this idea from the Infinite monkey theorem, which states that: "The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare." On that basis, that means that if the Universe really is infinite, then somewhere there must be a random number generator that keeps producing the same number over and over again simply because in an infinite Universe, however small the chance/probability is, a random number generator that produces the same number every time must exist. My head hurts :) Does anyone understand what I'm trying to explain?
I know you believe you understood what you think I said, but I am not sure you realize what you heard is not what I meant.
I would say that there could be a RNG that always has produced the same number every time it has been polled; but to say that it always does produce the same number would be a misnomer; it would no longer be a RNG if it had the property of always producing the same number. With a true RNG, perhaps it has always been observed to produce the same number; but each time you poll it, you always have the same probability of producing that number as of producing any other number (not, I would like to point out, less, as some might naively think). When you pull the lever the three millionth time and get the same number, it will be, for that pull, no more unusual than the first time and got that number, though you will definitely be in the presence of an entirely unprecedented run. No more unprecedented than any other three million digit string, however: you will have the same probability of matching any other predetermined three million digit string, or even having two RNGs produce two identical three million digit strings. Now, if only I had a Sports Almanac from the future for every digit that was going to come out of that RNG......
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Lloyd Atkinson☺ wrote:
My head hurts Does anyone understand what I'm trying to explain?
Sure, and somewhere, sometime, it's possible that someone will flip a coin 50 times and get heads every time. Now, take an aspirin and lay down.
Steve Wellens
Steve Wellens wrote:
Sure, and somewhere, sometime, it's possible that someone will flip a coin 50 times and get heads every time.
Try flipping a coin with "heads" face up, ensuring that it flips 4 times before it lands, what face is showing?
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Digital Thunder wrote:
where does one universe meet another (for there to be more than one universe there surely has to be a boundary and if there is a boundary they can't be infinite)
Oh sheesh. Consider two pieces of paper, each of infinite area. Now stack them, one on top of the other. Sure, they have a boundary - but are they any less infinite because of this fact? No, rather their boundary is perpendicular to their infinite axes, and so there is no issue. But why do they have to even have a boundary? Lift the top one of the other, and now there is no boundary: they are parallel, and there is a gap between them. Or is the gap the boundary? Okay then, lets remove one more dimension. Let's take two lines of infinite length. We could have them be parallel, so that they never meet; or skew lines, so that even the gap between them gets difficult to define. How about perpendicular skew lines? If one runs along the x-axis and the other along the y-axis, but they are separated along z-axis, then only one point is even really classifiable as close (not totally, but things are getting farther away from having a boundary all the time). Or we can add a dimension. Let's say infinite line A flashes in and out of existence, and the next day, infinite (skew) line B flashes in and out of existence. Even more removal from having a boundary, and still infinite. So what was your point exactly?
Digital Thunder wrote:
if space is infinite (to fit these infinite universes in) is the volume of space larger than the sum of its universes (so is one infinity bigger than another)?
Eh - I think someone else already covered this reasonably well. Yes, there are infinities that are larger than others - but this isn't one of them. It's exactly the same order of infinity.
Trevortni wrote:
Oh sheesh. Consider two pieces of paper, each of infinite area. Now stack them, one on top of the other. Sure, they have a boundary - but are they any less infinite because of this fact? No, rather their boundary is perpendicular to their infinite axes, and so there is no issue. But why do they have to even have a boundary? Lift the top one of the other, and now there is no boundary: they are parallel, and there is a gap between them. Or is the gap the boundary? Okay then, lets remove one more dimension. Let's take two lines of infinite length. We could have them be parallel, so that they never meet; or skew lines, so that even the gap between them gets difficult to define. How about perpendicular skew lines? If one runs along the x-axis and the other along the y-axis, but they are separated along z-axis, then only one point is even really classifiable as close (not totally, but things are getting farther away from having a boundary all the time). Or we can add a dimension. Let's say infinite line A flashes in and out of existence, and the next day, infinite (skew) line B flashes in and out of existence. Even more removal from having a boundary, and still infinite. So what was your point exactly?
Touching vaguely on M theory here... "Anyone for tennis?" Wait, I have to re-string my racquet... Pun intended.
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Your reasoning is incorrect. First, the universe is REALLY big, but it is not infinite. So there can only be a finite number of random generators that we have to work with. Second: The probability of getting N heads in a row decreases exponentially with N. The probability of getting infinite heads in a row is exactly zero. If we allow an infinite number of random generators, then we still get zero probability. This is by L'Hopital's rule: the probability of successive heads decreases toward zero more quickly than number of generators increase toward infinity. I did not have time to read the other comments. My humble apologies if I am repeating something that has already been said.
mutantdna wrote:
Second: The probability of getting N heads in a row decreases exponentially with N. The probability of getting infinite heads in a row is exactly zero.
Quite simply put... No. Read another reply I made in this thread, have a coin facing heads up, flip it, it turns head over tail exavtly 4 times (or 6 or 8 or 10 it doesn't matter). It always lands heads up, no matter how many times you flip it. In this case, the probablility of infinite heads in a row is exactly 100%. Sorry to burst your bubble.
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mutantdna wrote:
Second: The probability of getting N heads in a row decreases exponentially with N. The probability of getting infinite heads in a row is exactly zero.
Quite simply put... No. Read another reply I made in this thread, have a coin facing heads up, flip it, it turns head over tail exavtly 4 times (or 6 or 8 or 10 it doesn't matter). It always lands heads up, no matter how many times you flip it. In this case, the probablility of infinite heads in a row is exactly 100%. Sorry to burst your bubble.
I am afraid I do not understand your logic. Probability of 1 consecutive heads = 1/2. Probability of 2 consecutive heads = 1/4. Probability of 3 consecutive heads = 1/8. Probability of N consecutive heads = 1/(2**N) This to me is an exponential decrease. If you have an infinite number of coin flips, the chances of always having heads is zero. Lets try another way: For a finite number of coin flips, you get a binomial distribution of values with a standard deviation that is related to N. As N becomes arbitrarily large, the standard deviation goes to zero. This means that you will can ONLY expect a 50% heads for an infinite number of flips - and nothing else. This is an example of the central limit theorem if my memory serves me right. If you allow an infinite number of random generators, then this becomes a simple limit problem of the sort that is typically taught in first year university engineering or math courses. Am I missing something?
