5 ms·

"If the universe lasts forever, then any event that can happen, will happen, no matter how unlikely." That's just plain mathematically wrong. There are many ki

by zeteo 16y ago

"If the universe lasts forever, then any event that can happen, will happen, no matter how unlikely." That's just plain mathematically wrong. There are many kinds of infinity (http://en.wikipedia.org/wiki/Aleph_number http://en.wikipedia.org/wiki/Aleph_number), and the set of all possible events has a much, much higher cardinality than an infinite axis of time. This is similar to saying that, since the set of integers is infinite, it must contain Pi.

hugh3 16y ago
the set of all possible events has a much, much higher cardinality than an infinite axis of time. Are you sure of that? How would you prove it? It seems intuitively right, but intuition has a habit of being wrong, when it comes to infinities.
- jdavid 16y ago
  if they universe is composed of a single axis by which information is sorted, then infinite time should produce all possible combinations. However, if the universe has some 'structure' to it, it might become ordered at some point, exchanging chaos for order. In a universe that becomes ever more ordered and unique. This isn't like saying infinity - infinity, or infinity/infinity, this is more like there are a set of the infinite, and there is a set of the finite, and they are two distinct sets without one influencing the other. Order and Chaos need not be interchangeable. There is an interesting book called "Sync" which discusses the presence of spontaneous order from chaos. He also talks about how order can turn back into chaos, but the important part of the book is that there are cases where chaos will not turn into order, and order will not turn into chaos, meaning that the stability of the two create two distinct sets, one that contains the identity of the universe and one that allows it to change. Some of you may have read Wolframs work as well which shows that simple rules can lead to ordered stable systems. That stability is the definition of uniqueness. Sync, it's the one book I recommend all my thinking friends read. http://amzn.to/a8s5Dl http://amzn.to/a8s5Dl It changed my life, and my world view. So, I am comfortable saying that no act of randomness by any number of chimpanzees ( as they are now ) will randomly produce a work of William Shakespeare. WS, wasn't just a man, he was a man in a time with billions of events that influenced that moment when he composed those works. There was a structure and a path to those influences, and it wasn't such a frail construct that randomness happened upon it, but rather there was a beautify symmetry that occurred and allowed that to come to pass. Another way to look at it, is that random/ chaotic signals will not sum up to collapse a bridge. however an ordered marching army has in many cased taken down the bridge.
- Robin_Message 16y ago
  I have a machine that flashes a light on or off once a second, at random. You can view streams of output from the machine as possible events. Now, suppose an infinite amount of time has passed (aleph-null, by definition). Have I seen every possible sequence? Well, take the set of aleph-null length sequences (all aleph-null of them [1]) and put them in an order. Now construct the following sequence: invert bit 1 from sequence 1, invert bit 2 from sequence 2, ... etc. Note that this sequence was never produced by the machine, since we enumerated all the sequences it produced, but they are all different to this one. So, by a standard diagonalisation, the set of events is higher cardinality. [1] How many aleph-null length sequences come out of the machine? Well, they have to be continuous, so the only obvious aleph-null length sequence is the total sequence the machine produces. However, you can drop an entry from the beginning of the sequence to get another aleph-null length sequence. And so on, making a total of aleph-null.
  - hugh3 16y ago
    Interesting attempt, but I'm not convinced by the part of the argument where you divide the complete output of the flashy light box into aleph-null sequences, each of length aleph-null. Mostly, I'm unconvinced that an "event" which takes an infinitely long time to complete actually counts as an event. Usually when we talk about events they're localised in space and time.
    - theli0nheart 16y ago
      Well, it's kind of a Catch-22. This proof is dependent upon the lemma that time is infinite. On the other hand, the set of events we're looking for is the one where time ends. Therefore by assuming the lemma we have no need for the proof. OTOH, if we're looking at another event that's not so confusing (one that's not the end of time), the proof holds water. It's a pretty sweet proof that I think is similarly used to prove that the set of real numbers is uncountable.
      - hugh3 16y ago
        No, what I'm saying is that an event which takes an infinitely long time to happen (ie an infinite series of one-second flashes) isn't a proper "event", which as used at least in relativity, is something that happens in finite space and time. But there's no need to argue about the definition of event. I'll say instead that I'm interested in knowing whether infinite time implies that all finite-length events must eventually happen (and indeed, happen an infinite number of times as the paper claims).
- j_baker 16y ago
  Events happen at the same time. At any one given point in time, there are trillions (probably much more than that) of events happening. In fact, wouldn't the set of all events be uncountably infinite? Now that I think about it, we're talking about all possible events, which is even bigger. For any given event, there is theoretically an infinite amount of alternative possible events. Therefore, for each of the trillions of events at any given point in time, there is an infinite set of other possible events.
  - hugh3 16y ago
    Certainly there's an uncountably infinite number of possible events. The question is whether that's an uncountable infinity with larger cardinality than the uncountable infinity of instants in which they could be happening. As far as I'm concerned having read the thread up to this point the question is still open... though my personal prejudice is still that it's not true that every event must happen.
Sam_Odio 16y ago
> This is similar to saying that, since the set of integers is infinite, it must contain Pi. Not exactly. Pi has no chance of occurring in an infinite set of integers.
- someone_here 16y ago
  Okay, so how about "Given a number generator that produces numbers for an infinite amount of time, it will produce Pi" This, of course, is impossible.
  - twymer 16y ago
    I think you're missing the point. While it's _extremely_ unlikely that the next random number produced would be Pi, in an infinite scale it can (and theoretically would) happen because it is, in fact, possible to occur.
    - yummyfajitas 16y ago
      No, it theoretically would not happen. Proof: let X[n] be the set of numbers with non-zero probability of being produced at the n'th trial. X[n] must be countable, since sum(X[n]) = 1 and the sum of any uncountably infinite set of non-zero numbers must be infinite. Let X = union(X[n], n=0...infinity). X is countable, being the countable union of countable sets. The reals are uncountable. Thus, most real numbers will NOT eventually be produced. (It's true, pi in particular could be in X, but the vast majority of numbers could not be in X.)
      - eru 16y ago
        By a similar argument, at least half of all natural numbers are completely random.
      - twymer 16y ago
        I'm not entirely sure I understand the logic here, and being not a math student myself I'm sure my original statement is likely to be wrong. However, I don't see how the set of numbers found in an infinite number of trials would be countable. Edit: I googled some about this and can say I definitely learned some math today..
asmithmd1 16y ago
Well, there IS an integer that contains all the digits of Pi 3141592...
- hugh3 16y ago
  No, that "integer" would need to have an infinite number of digits, and no integer has an infinite number of digits.
  - asmithmd1 16y ago
    So you are saying that their is a largest integer with a finite number of digits? Lets call that integer N, wouldn't 10*N have one more digit than N?
    - crazyjimbo 16y ago
      Integers can have an arbitrarily large but finite number of digits. They cannot have an infinite number of digits. There is a big difference.
      - asmithmd1 16y ago
        I guess we are in the realm of cardinality of infinity which I don't quite get. I would think an infinitely big integer would have an infinite number of digits. So you are saying: log(infinity) < infinity
        hugh3 16y ago
        Once again, there is no such thing as an infinitely big integer. You seem to be having difficulty with the fact that: 1. There are an infinite number of integers, but 2. Every integer is finite Start counting 1,2,3,4... Do you ever get to the end? No, this proves that there are infinitely many integers. But do you ever get to an infinite number? No, you'll stay in the set of finite integers forever.
    - hugh3 16y ago
crazyjimbo 16y ago
Their statement makes sense: If you have a non zero probability of something happening, say P, and N times that it could happen, then you would expect it to happen about P * N times. Lets let E = P * N. Now as N goes to infinity the number of times you expect it to happen does as well. Both go to a countable infinity (Aleph 0 as you linked) which is a problem when it comes back to defining P. Normally we would say P = E / N but this doesn't make sense when E and N are both the same 'size' of infinity.
alphaBetaGamma 16y ago
To give a specific example: In three and more dimensions, if you start a random walk outside some sphere there is a non zero probability that the event "the brownian motion enters this sphere" never happens. Funnily, in dimension one and two this is not true.
- hugh3 16y ago
  Interesting point, and what's interesting is that it doesn't rely on the cardinality arguments above. Both the set of points visited by the walker and the set of points in the universe have the same cardinality, but that doesn't mean that the walker visits all points. (In fact I suspect he can never visit more than a vanishing subset of them, but I could be wrong, and I'm sure somebody has figured this out already...)
- jules 16y ago
  Interesting. Are you talking about a continuous walk, or a walk on the integers? Is there an intuitive explanation for this? I get that in 1 dimension you will visit all points if you wait long enough, but two? And why does it change at 3?
  - alphaBetaGamma 16y ago
    This is for continuous walk. As you say, in one dimension you will visit all the points. In two dimension the random walk is dense, so you pass arbitrarily close to every point in the plane. As the dimension increases beyond two, it gets less and less likely that you hit the target sphere. In some way, space gets bigger and bigger as the dimension increases. Suppose the d dimensional "target" sphere is very small and very close to the starting point of the path. Let some time pass, and suppose that the Brownian motion does not hit the sphere and has increased by 1 in every direction. Then the distance to the sphere is sqrt(d), which grows unbounded as d increases: if you don't hit the target immediately, on average you will be further away from it in higher dimension. So the probability of touching the target after that diminishes as the dimension increases. I don't have an intuitive explanation of why 3 is the critical dimension though.
ohyes 16y ago
This does not seem wrong to me. "Among other things it depends crucially on an important assumption about the laws of physics: that we ought to be able to understand why they work, not just observe that they do work" IANAP but, In assuming that there are discrete laws of physics, we also assume there is a subset of all 'possible' events; events that 'can happen'. (Given a certain arrangement of the universe at T1, there is a set of possible arrangements of the universe at T2). If we assume finite time, the set of events that will happen is a subset of the set of events that can happen. This is because the set of events that can happen is infinite and the set of events that will happen is finite. If both are infinite, then they are the same set. Infinite time makes it apparent that all of the events that can happen will happen an infinite number of times, because events that can happen are events that will happen, and an event that can happen is never eliminated from the set of events that can happen by having been an event that will happen.
- mikemol 16y ago
  "In assuming that there are discrete laws of physics, we also assume there is a subset of all 'possible' events; events that 'can happen'. (Given a certain arrangement of the universe at T1, there is a set of possible arrangements of the universe at T2)." False. Possibilities can be mutually exclusive, so if one occurs, the other can't. Yet, after the fact, we considered them both to be events that could have happened. (Unless you subscribe to absolute determinism, but I don't believe that's clear to be the case. Also, IME, discussions of determinism mechanisms in, e.g. quantum mechanics, tend to talk more about whether that's where Diety lives.)

by zeteo 16y ago

hugh3 16y ago

the set of all possible events has a much, much higher cardinality than an infinite axis of time. Are you sure of that? How would you prove it? It seems intuitively right, but intuition has a habit of being wrong, when it comes to infinities.

jdavid 16y ago
if they universe is composed of a single axis by which information is sorted, then infinite time should produce all possible combinations. However, if the universe has some 'structure' to it, it might become ordered at some point, exchanging chaos for order. In a universe that becomes ever more ordered and unique. This isn't like saying infinity - infinity, or infinity/infinity, this is more like there are a set of the infinite, and there is a set of the finite, and they are two distinct sets without one influencing the other. Order and Chaos need not be interchangeable. There is an interesting book called "Sync" which discusses the presence of spontaneous order from chaos. He also talks about how order can turn back into chaos, but the important part of the book is that there are cases where chaos will not turn into order, and order will not turn into chaos, meaning that the stability of the two create two distinct sets, one that contains the identity of the universe and one that allows it to change. Some of you may have read Wolframs work as well which shows that simple rules can lead to ordered stable systems. That stability is the definition of uniqueness. Sync, it's the one book I recommend all my thinking friends read. http://amzn.to/a8s5Dl http://amzn.to/a8s5Dl It changed my life, and my world view. So, I am comfortable saying that no act of randomness by any number of chimpanzees ( as they are now ) will randomly produce a work of William Shakespeare. WS, wasn't just a man, he was a man in a time with billions of events that influenced that moment when he composed those works. There was a structure and a path to those influences, and it wasn't such a frail construct that randomness happened upon it, but rather there was a beautify symmetry that occurred and allowed that to come to pass. Another way to look at it, is that random/ chaotic signals will not sum up to collapse a bridge. however an ordered marching army has in many cased taken down the bridge.
Robin_Message 16y ago
I have a machine that flashes a light on or off once a second, at random. You can view streams of output from the machine as possible events. Now, suppose an infinite amount of time has passed (aleph-null, by definition). Have I seen every possible sequence? Well, take the set of aleph-null length sequences (all aleph-null of them [1]) and put them in an order. Now construct the following sequence: invert bit 1 from sequence 1, invert bit 2 from sequence 2, ... etc. Note that this sequence was never produced by the machine, since we enumerated all the sequences it produced, but they are all different to this one. So, by a standard diagonalisation, the set of events is higher cardinality. [1] How many aleph-null length sequences come out of the machine? Well, they have to be continuous, so the only obvious aleph-null length sequence is the total sequence the machine produces. However, you can drop an entry from the beginning of the sequence to get another aleph-null length sequence. And so on, making a total of aleph-null.
- hugh3 16y ago
  Interesting attempt, but I'm not convinced by the part of the argument where you divide the complete output of the flashy light box into aleph-null sequences, each of length aleph-null. Mostly, I'm unconvinced that an "event" which takes an infinitely long time to complete actually counts as an event. Usually when we talk about events they're localised in space and time.
  - theli0nheart 16y ago
    Well, it's kind of a Catch-22. This proof is dependent upon the lemma that time is infinite. On the other hand, the set of events we're looking for is the one where time ends. Therefore by assuming the lemma we have no need for the proof. OTOH, if we're looking at another event that's not so confusing (one that's not the end of time), the proof holds water. It's a pretty sweet proof that I think is similarly used to prove that the set of real numbers is uncountable.
    - hugh3 16y ago
      No, what I'm saying is that an event which takes an infinitely long time to happen (ie an infinite series of one-second flashes) isn't a proper "event", which as used at least in relativity, is something that happens in finite space and time. But there's no need to argue about the definition of event. I'll say instead that I'm interested in knowing whether infinite time implies that all finite-length events must eventually happen (and indeed, happen an infinite number of times as the paper claims).
j_baker 16y ago
Events happen at the same time. At any one given point in time, there are trillions (probably much more than that) of events happening. In fact, wouldn't the set of all events be uncountably infinite? Now that I think about it, we're talking about all possible events, which is even bigger. For any given event, there is theoretically an infinite amount of alternative possible events. Therefore, for each of the trillions of events at any given point in time, there is an infinite set of other possible events.
- hugh3 16y ago
  Certainly there's an uncountably infinite number of possible events. The question is whether that's an uncountable infinity with larger cardinality than the uncountable infinity of instants in which they could be happening. As far as I'm concerned having read the thread up to this point the question is still open... though my personal prejudice is still that it's not true that every event must happen.

Sam_Odio 16y ago

> This is similar to saying that, since the set of integers is infinite, it must contain Pi. Not exactly. Pi has no chance of occurring in an infinite set of integers.

someone_here 16y ago
Okay, so how about "Given a number generator that produces numbers for an infinite amount of time, it will produce Pi" This, of course, is impossible.
- twymer 16y ago
  I think you're missing the point. While it's _extremely_ unlikely that the next random number produced would be Pi, in an infinite scale it can (and theoretically would) happen because it is, in fact, possible to occur.
  - yummyfajitas 16y ago
    No, it theoretically would not happen. Proof: let X[n] be the set of numbers with non-zero probability of being produced at the n'th trial. X[n] must be countable, since sum(X[n]) = 1 and the sum of any uncountably infinite set of non-zero numbers must be infinite. Let X = union(X[n], n=0...infinity). X is countable, being the countable union of countable sets. The reals are uncountable. Thus, most real numbers will NOT eventually be produced. (It's true, pi in particular could be in X, but the vast majority of numbers could not be in X.)
    - eru 16y ago
      By a similar argument, at least half of all natural numbers are completely random.
    - twymer 16y ago
      I'm not entirely sure I understand the logic here, and being not a math student myself I'm sure my original statement is likely to be wrong. However, I don't see how the set of numbers found in an infinite number of trials would be countable. Edit: I googled some about this and can say I definitely learned some math today..

asmithmd1 16y ago

Well, there IS an integer that contains all the digits of Pi 3141592...

hugh3 16y ago
No, that "integer" would need to have an infinite number of digits, and no integer has an infinite number of digits.
- asmithmd1 16y ago
  So you are saying that their is a largest integer with a finite number of digits? Lets call that integer N, wouldn't 10*N have one more digit than N?
  - crazyjimbo 16y ago
    Integers can have an arbitrarily large but finite number of digits. They cannot have an infinite number of digits. There is a big difference.
    - asmithmd1 16y ago
      I guess we are in the realm of cardinality of infinity which I don't quite get. I would think an infinitely big integer would have an infinite number of digits. So you are saying: log(infinity) < infinity
      - hugh3 16y ago
        Once again, there is no such thing as an infinitely big integer. You seem to be having difficulty with the fact that: 1. There are an infinite number of integers, but 2. Every integer is finite Start counting 1,2,3,4... Do you ever get to the end? No, this proves that there are infinitely many integers. But do you ever get to an infinite number? No, you'll stay in the set of finite integers forever.
  - hugh3 16y ago

crazyjimbo 16y ago

Their statement makes sense: If you have a non zero probability of something happening, say P, and N times that it could happen, then you would expect it to happen about P * N times. Lets let E = P * N. Now as N goes to infinity the number of times you expect it to happen does as well. Both go to a countable infinity (Aleph 0 as you linked) which is a problem when it comes back to defining P. Normally we would say P = E / N but this doesn't make sense when E and N are both the same 'size' of infinity.

alphaBetaGamma 16y ago

To give a specific example: In three and more dimensions, if you start a random walk outside some sphere there is a non zero probability that the event "the brownian motion enters this sphere" never happens. Funnily, in dimension one and two this is not true.

hugh3 16y ago
Interesting point, and what's interesting is that it doesn't rely on the cardinality arguments above. Both the set of points visited by the walker and the set of points in the universe have the same cardinality, but that doesn't mean that the walker visits all points. (In fact I suspect he can never visit more than a vanishing subset of them, but I could be wrong, and I'm sure somebody has figured this out already...)
jules 16y ago
Interesting. Are you talking about a continuous walk, or a walk on the integers? Is there an intuitive explanation for this? I get that in 1 dimension you will visit all points if you wait long enough, but two? And why does it change at 3?
- alphaBetaGamma 16y ago
  This is for continuous walk. As you say, in one dimension you will visit all the points. In two dimension the random walk is dense, so you pass arbitrarily close to every point in the plane. As the dimension increases beyond two, it gets less and less likely that you hit the target sphere. In some way, space gets bigger and bigger as the dimension increases. Suppose the d dimensional "target" sphere is very small and very close to the starting point of the path. Let some time pass, and suppose that the Brownian motion does not hit the sphere and has increased by 1 in every direction. Then the distance to the sphere is sqrt(d), which grows unbounded as d increases: if you don't hit the target immediately, on average you will be further away from it in higher dimension. So the probability of touching the target after that diminishes as the dimension increases. I don't have an intuitive explanation of why 3 is the critical dimension though.

ohyes 16y ago

This does not seem wrong to me. "Among other things it depends crucially on an important assumption about the laws of physics: that we ought to be able to understand why they work, not just observe that they do work" IANAP but, In assuming that there are discrete laws of physics, we also assume there is a subset of all 'possible' events; events that 'can happen'. (Given a certain arrangement of the universe at T1, there is a set of possible arrangements of the universe at T2). If we assume finite time, the set of events that will happen is a subset of the set of events that can happen. This is because the set of events that can happen is infinite and the set of events that will happen is finite. If both are infinite, then they are the same set. Infinite time makes it apparent that all of the events that can happen will happen an infinite number of times, because events that can happen are events that will happen, and an event that can happen is never eliminated from the set of events that can happen by having been an event that will happen.

mikemol 16y ago
"In assuming that there are discrete laws of physics, we also assume there is a subset of all 'possible' events; events that 'can happen'. (Given a certain arrangement of the universe at T1, there is a set of possible arrangements of the universe at T2)." False. Possibilities can be mutually exclusive, so if one occurs, the other can't. Yet, after the fact, we considered them both to be events that could have happened. (Unless you subscribe to absolute determinism, but I don't believe that's clear to be the case. Also, IME, discussions of determinism mechanisms in, e.g. quantum mechanics, tend to talk more about whether that's where Diety lives.)