View Full Version : In an infinite universe must even impossible events happen at least once?

marsbug

2008-Apr-28, 03:48 PM

My maths extends only as far as basic calulus, so what i'm about to ask may be based upon bad logic but: If an event has a one in infinity chance of occuring, this may be seen as equivelent to a zero chance of occuring (or at least infinitely close to it). Hence if an event has zero chance of occuring, and the universe is infinitely large, does this mean it must happen once? :liar::lol:

P.S :before you all start yelling I am running a fever at the moment....:sick:

tommac

2008-Apr-28, 03:54 PM

My maths extends only as far as basic calulus, so what i'm about to ask may be based upon bad logic but: If an event has a one in infinity chance of occuring, this may be seen as equivelent to a zero chance of occuring (or at least infinitely close to it). Hence if an event has zero chance of occuring, and the universe is infinitely large, does this mean it must happen once? :liar::lol:

P.S :before you all start yelling I am running a fever at the moment....:sick:

no ... by definition it is impossible. Now even very unlikely situations would happen.

grant hutchison

2008-Apr-28, 04:12 PM

My maths extends only as far as basic calulus, so what i'm about to ask may be based upon bad logic but: If an event has a one in infinity chance of occuring, this may be seen as equivelent to a zero chance of occuring (or at least infinitely close to it).I think the mistake here is to think that "one chance in infinity" actually means something useful.

An impossible event has zero probability. Things with zero probability won't ever happen.

Grant Hutchison

John Mendenhall

2008-Apr-29, 07:20 PM

I think the mistake here is to think that "one chance in infinity" actually means something useful.

An impossible event has zero probability. Things with zero probability won't ever happen.

Grant Hutchison

As I recall, Grant and I kicked this around once before. If I remember correctly, Grant's point was that even though an event may have a probability > 0, there may not be enough time available in the universe for it to be likely to happen. For example, the probability of a macroscopic object popping into existence is quantum mechanically possible, but fantastically improbable. And there are other examples. But if an event is impossible, like intermediate electron shells, it will never happen, even in an infinite amount of time.

mugaliens

2008-Apr-29, 07:38 PM

My maths extends only as far as basic calulus, so what i'm about to ask may be based upon bad logic but: If an event has a one in infinity chance of occuring, this may be seen as equivelent to a zero chance of occuring (or at least infinitely close to it). Hence if an event has zero chance of occuring, and the universe is infinitely large, does this mean it must happen once? :liar::lol:

P.S :before you all start yelling I am running a fever at the moment....:sick:

Statistically speaking, you're talking about a condition where both the numerator and the denominator are approaching zero.

So what's the answer?

Well, that depends upon which is approaching zero faster!

Unfortunately, with English, the answer is indetermanent. Even in the pure language of math I think that for me, it would still be indetermanent! You might try some of the others on this board, IF you can couch your question in the pure language of math...

BISMARCK

2008-Apr-29, 09:46 PM

I would think that in a universe with truly infinite time, any event with a probability greater than zero must happen eventually.

Senor Molinero

2008-Apr-30, 06:44 AM

The key word there Bismarck is eventually. I may actually never witness my rolling a 6 on a dice, though the probability be very small of this happening, it will happen eventually. One could roll a dice for a billion years and never see a 6. Eventually includes never. That being after an infinite length of time has passed.

Jens

2008-Apr-30, 07:36 AM

Statistically speaking, you're talking about a condition where both the numerator and the denominator are approaching zero.

I think this is a reasonable interpretation. I think there are three possibililies.

(1) The event has no chance of occurring. In this case, the universe can go on forever and the event will never happen. This is true by definition, since zero probability means that it will never happen forever and ever.

(2) The event has a finite chance of happening (like rolling 1 on a dice). In this case, it will happen eventually.

(3) The problem is an event that has an infinitely small chance of happening. In this case, I don't know what would happen.

I brought this last case up in the past in a thread on Zeno's paradox. I was wondering what whether you could get from A to B, even if there are infinite points between them, if there is also an infinity of time points between now and a certain time. It's like the problem of whether an inifinite number of guests would fill up an infinitely large hotel. I doubt it's answerable, but I'm not a mathematican.

Jeff Root

2008-Apr-30, 08:54 AM

What distinguishes between an event being possible or being impossible,

if it has never been observed to happen? It must certainly be possible for

a snowflake to form from exactly ten trillion water molecules, even though

(I think it is safe to say) such a snowflake has never had its molecules

counted precisely. But would it be possible for a mountainside to form an

exact replica of the sculpture on Mount Rushmore, without the intermediate

steps of living creatures deliberately planning and executing it? Assuming

that an exact replica of Mount Rushmore could form naturally in the first

place, it would then be necessary for bits of the mountain to fall off in just

the right places to duplicate the sculpture. How could anyone determine

whether that would be possible?

-- Jeff, in Minneapolis

Jens

2008-Apr-30, 09:16 AM

What distinguishes between an event being possible or being impossible,

if it has never been observed to happen?

When I was thinking of things that are impossible, I was thinking of things that can't happen by definition. For example, what are the odds of us finding a star that contains oxygen but doesn't contain oxygen. Excluding word games, I think it would be an impossibility logically. I don't think you could ever have a chance in the laws of logic, so I would classify this as known to be impossible. Or for example, for a person to be born on the day before the day he was born (according to regular definitions of the words, not messing with time zones or things like that).

Jeff Root

2008-Apr-30, 09:39 AM

Jens,

I wasn't thinking about cases like that, because they are trivial, and

automatically go into the "impossible" category without need of further

consideration. John Mendenhall's suggestion of "intermediate electron

shells" would be an example of something that requires thinking about

before putting it into the "impossible" category. My Mount Rushmore

sculpture example seems impossible to categorize.

-- Jeff, in Minneapolis

Ivan Viehoff

2008-Apr-30, 11:09 AM

I think the mistake here is to think that "one chance in infinity" actually means something useful. An impossible event has zero probability. Things with zero probability won't ever happen.

Something has to happen. We can easily construct a model where every individual possibility has zero probability, eg, pick a number on the real line, by spinning a pin, but one of them happens. All right the real world is imperfectly modelled by euclidean space. But if we are going to use probability as a model, then we have to acknowledge that in any model of continuous probability, things of zero probability do happen.

But on the original question. An infinite set can be very "sparse" in comparison to an infinite set that contains it. Eg, the function that is 1 at all rational points and 0 at all irrational points has an integral of zero - it is a null set, a set of measure zero, even though it is dense in the continuum (ie, its topological closure is the continuum, in particular, for every point in the continuum we can find a rational which is arbitrarily close to it.) In this case the set of the rationals has smaller cardinality than the continuum (there are different "sizes" of infinity, in a sense that can be made precise, which are called cardinalities, so we can say that there are some infinities which really are bigger than other infinities, much, much, infinitely, bigger). So the fact that they are of measure zero is unsurprising. But even a set of the same cardinality as the continuum can have zero measure, eg, the Cantor set is also null/of zero measure. Even more amazingly the Cantor set is nowhere dense - it is a closed set, we cannot find points in it that are arbitrarily close to points outside it. The rationals, although of smaller cardinality than the Cantor set, look like they are everywhere; but the Cantor set (albeit in 2d) is a bit like a cheese where the mice have enlarged the holes until there is nothing but a vanishingly thin skin between the holes, and in effect the remaining cheese is all hole.

I suspect the number of possibilities of what could happen is in general going to be not just larger than a description of what actually happens in an infinite universe, but of larger cardinality. So you can have an infinite universe, and the things that do happen are likely to be a very small/sparse set in comparison to the things that could happen. Somewhere else in an infinite universe, we do not have to propose that there is also a precise copy of the earth where everything is exactly the same except that I am a rich, brilliant, talented, perfectly empathetic and a stunning hunk. There are enough things to fill an infinite universe without having anything else very much like the earth and its precise history.

Oh yeah, and impossible things don't happen. They are excluded from happening.

phunk

2008-Apr-30, 03:54 PM

This is a very simple question. If they are impossible, then the answer is no, by definition. Otherwise, they are not impossible, just improbable.

tommac

2008-May-01, 03:50 AM

Firstly, if there is even a remote possibility of something happening ... in an infinite amount of time it not only can happen but it will happen.

Now for a slight off subject but related question.

If there is any chance for a door opening either to a point backwards in time or to another universe even if improbable it will happen in an infinite amount of time.

Althought I dont think that time travel back in time would be possible if at the speed of light you could travel to the point in which you left. Doing so would put you in a position of the gradfathers paradox and reset all quantum probability.

alainprice

2008-May-01, 04:23 AM

Some 'things' will interfere with other 'things'.

If we look at our own universe, which ages, matures, and eventually might die, there will be finite probabilities.

agingjb

2008-May-01, 07:33 AM

Obviously if an event is impossible, then by definition it won't happen.

But would every possible event take place in an infinite universe? You might think so, but take it another step: in all the infinite number of possible infinite universes there must be an infinite number where some possible events just happen not to occur.

WaxRubiks

2008-May-01, 07:45 AM

if something has a non-zero chance of happening in a universe that is infinite and with infinite time, then not only will it happen, it will happen an infinite number of times.

Jens

2008-May-01, 11:28 AM

Now for a slight off subject but related question.

If there is any chance for a door opening either to a point backwards in time or to another universe even if improbable it will happen in an infinite amount of time.

It comes down to whether the possibility exists or not. For example, if there are no other universes, then it would be impossible no matter how long you wait. If time travel is impossible, then likewise.

Ivan Viehoff

2008-May-01, 12:58 PM

if something has a non-zero chance of happening in a universe that is infinite and with infinite time, then not only will it happen, it will happen an infinite number of times.

Not so. The train leaving Euston at 0900 on 2 May 2008 has, according to experience, about a 15% chance of being late. But once the 0900 train on 2 May 2008 has happened, then it either it was late or it wasn't late, and the 0900 train on 2 May 2008 will never happen again; on another day I'll be older, the relative location of Jupiter (for example) will be different, etc, so it is not the same event.

Every state of the universe that can be attained must be attained in infinite time? Not so.

One way to think of it: An infinite set can have a strict subset that is also infinite (think of square numbers as a subset of integers, both infinite but the former much more sparse). So an infinite universe can still just visit a small subset of the set of all possible states, and still be infinite without having visited every possible state on the way.

But actually I think that the size of infinity which is the description of all the states of the universe that can be attained is a larger infinity than all the states of the universe that are attained in an infinite time. So the universe cannot even possibly pass through all possible states it could attain, even if the universe is infinite in extent and present for infinite time, because the number of possible states is too large. This is like saying you cannot, even in principle, make a list of the real numbers and guarantee that you would eventually visit every one, even in infinite time. In particular I point to the fact that if X is a set then the cardinality of 2X is strictly larger, even for infinite sets.

I suggest, if you are unfamiliar, investigating the concept of different sizes of infinity.

http://en.wikipedia.org/wiki/Countable

http://en.wikipedia.org/wiki/Uncountable_set

http://en.wikipedia.org/wiki/Cardinal_number

Eroica

2008-May-01, 02:51 PM

... if an event has zero chance of occuring, and the universe is infinitely large, does this mean it must happen once? :liar::lol:

You know how they say that parallel lines meet at (spatial) infinity? Could you, then, say that impossible events are ones that occur at (temporal) infinity...? :think:

Disinfo Agent

2008-May-01, 02:57 PM

My maths extends only as far as basic calulus, so what i'm about to ask may be based upon bad logic but: If an event has a one in infinity chance of occuring, this may be seen as equivelent to a zero chance of occuring (or at least infinitely close to it). Hence if an event has zero chance of occuring, and the universe is infinitely large, does this mean it must happen once? :liar::lol:You are talking about a zero probability event. It does not matter whether you say "a 1 in infinity chance", "a 2 in infinity chance", or "an n in infinity change". They all reduce to zero.

Impossible events have zero probability, but the converse is not true: events with zero probability can nevertheless happen. They're just extremely rare. On the other hand, being possible doesn't necessarily equal being observed.

Cougar

2008-May-01, 03:08 PM

Well, in quantum mechanics, I believe it was Feynman who said if it's not forbidden, it's compulsory.

Which is a little odd, since in QM, even things that are forbidden occur once in a while (tunneling).

As for the OP, you'll want to read Many Worlds in One [2006] by Alex Vilenkin. Like frog march, Vilenkin posits that "if something has a non-zero chance of happening in a universe that is infinite and with infinite time, then not only will it happen, it will happen an infinite number of times."

Oh, wait. Vilenkin is talking about an infinite number of "pocket" universes within an eternally inflating sea of cosmic inflation. Bringing "infinity" into the equation allows one to say there is another pocket universe out there (forever undetectable) with an exact copy of Cougar and frog march and Alain and Jens, et al., typing at their keyboards just like we are. Actually, an infinite number of them.

That's one reason I'm not too interested in pondering scenarios where infinity plays a part.....

tommac

2008-May-01, 04:20 PM

It comes down to whether the possibility exists or not. For example, if there are no other universes, then it would be impossible no matter how long you wait. If time travel is impossible, then likewise.

But if it is possible ( and the universe is infinite ) then it DOES happen ...

Alternatively

if the universe is NOT infinite but it is possible then eventually it will happen.

tommac

2008-May-01, 04:22 PM

Lloyd: What are the chances of a guy like you and a girl like me... ending up together?

Mary: Well, that's pretty difficult to say.

Lloyd: Hit me with it! I've come a long way to see you, Mary. The least you can do is level with me. What are my chances?

Mary: Not good.

Lloyd: You mean, not good like one out of a hundred?

Mary: I'd say more like one out of a million.

[pause]

Lloyd: So you're telling me there's a chance.

--dumb and dumber

Ivan Viehoff

2008-May-01, 04:24 PM

But if it is possible ( and the universe is infinite ) then it DOES happen ...

Not so. See my comment above when I quoted Frogmarch saying the same thing.

mugaliens

2008-May-01, 04:52 PM

My maths extends only as far as basic calulus, so what i'm about to ask may be based upon bad logic but: If an event has a one in infinity chance of occuring, this may be seen as equivelent to a zero chance of occuring (or at least infinitely close to it). Hence if an event has zero chance of occuring, and the universe is infinitely large, does this mean it must happen once? :liar::lol:

P.S :before you all start yelling I am running a fever at the moment....:sick:

The notation "-->" stands for "approaches."

As n-->infinity, 1/n-->0.

As n-->infinity, n/1-->infinity

As n-->infinity, n/n remains equal to 1.

Thus, if the chance of something occurring is 1/n, and n=infinity, given an infinate universe, the probability of that event occurring is 100%, as n/n is 1.

You have to be careful with this, though. For example, you couldn't say, "what's the liklihood of, say, John, the nine-year-old kid down the street, jumping to the Moon?"

Rather, you'd have to say, "what's the chance that there's a life-form out there that posesses enough wherewithall to jump (or fly without artificial means) to the moon?"

I've little doubt that if there is life on other planets, there is some life form out there that evolved rocket propulsion with sufficient thrust to get off an Earth-like planet and make it to it's own Moon-like moon. We see similarly improbably things here on Earth, including flying fish, termites that shoot poison out a nozzle on their forheads, and a certain primate with enough smarts to figure out an artificial means of getting off his planet, travelling a quarter of a million miles away, and landing on another celestial body.

And still have enough get-em-up-and-go juice to actually return to his home world!

grant hutchison

2008-May-01, 06:49 PM

We can easily construct a model where every individual possibility has zero probability, eg, pick a number on the real line, by spinning a pin, but one of them happens. All right the real world is imperfectly modelled by euclidean space. But if we are going to use probability as a model, then we have to acknowledge that in any model of continuous probability, things of zero probability do happen.

Impossible events have zero probability, but the converse is not true: events with zero probability can nevertheless happen.This is a way of thinking I hadn't previously acquired.

Thanks, both. :)

Grant Hutchison

agingjb

2008-May-01, 07:21 PM

Hmm. I won't comment on whether an event that does occurs can have zero probability; I'm rather unsure about measure of probabilities in an infinite domain. But I do wonder what, if any, mathematical abstraction is being to model infinite time: the real line, or some other construct?

I'd say that once we introduce real infinities as a hypothesis into the physical world, then we do get some tricky questions. One plausible idea is that everything not actually impossible happens (and infinitely often). Fair enough, but infinities tend to produce some very tricky logic (as has been pointed out).

Just one question: is the "time" in this infinite universe Archimedean, that is to say are every two instants in this infinite span a finite (but of course possibly very large) distance apart?

tommac

2008-May-01, 07:32 PM

Well this depends on the wording of the problem. We cant have a time dependency in the problem if time is the thing that stretches to infinity.

if you were to say that every atom in the universe would bounce into each other even if this is so remotely unlikely ( assuming physics allows this ) it will happen in a infinite amount of time.

However if you are saying that the 9:00 train will be late and then the 10:00 train will be on time ... because there is a time limitation involved. Then the fact that time goes to infinity is irrelevant.

The quote you are giving about infinite numbers is best discussed in a cool book called 1 2 3 infinity ... which discusses how some infinities cancel each other out. Something like the set of all integers vs the set of whole numbers. For each integer there is a set of infinite real numbers so it is infinity * infinity. The set of all possible 3d curves is another level up from there.

So the infinity we are talking about is a state of the universe at any given time ( I think you can have a series of times as long as they arent limited to a starting point ) that state WILL happen given enough time.

This being said ... the amount of energy in the universe may depleat. so there is a time variable on some things. It may be remotely possible now to have a huge explosion by 50 black holes simultaneously tearing each other apart ( I dont know if it would cause an explosion ) however the life of all black holes may be a finite number of years so there is no infinite term unless you also have an infinitely large universe. then you have two infinite terms.

Not so. The train leaving Euston at 0900 on 2 May 2008 has, according to experience, about a 15% chance of being late. But once the 0900 train on 2 May 2008 has happened, then it either it was late or it wasn't late, and the 0900 train on 2 May 2008 will never happen again; on another day I'll be older, the relative location of Jupiter (for example) will be different, etc, so it is not the same event.

Every state of the universe that can be attained must be attained in infinite time? Not so.

One way to think of it: An infinite set can have a strict subset that is also infinite (think of square numbers as a subset of integers, both infinite but the former much more sparse). So an infinite universe can still just visit a small subset of the set of all possible states, and still be infinite without having visited every possible state on the way.

But actually I think that the size of infinity which is the description of all the states of the universe that can be attained is a larger infinity than all the states of the universe that are attained in an infinite time. So the universe cannot even possibly pass through all possible states it could attain, even if the universe is infinite in extent and present for infinite time, because the number of possible states is too large. This is like saying you cannot, even in principle, make a list of the real numbers and guarantee that you would eventually visit every one, even in infinite time. In particular I point to the fact that if X is a set then the cardinality of 2X is strictly larger, even for infinite sets.

I suggest, if you are unfamiliar, investigating the concept of different sizes of infinity.

http://en.wikipedia.org/wiki/Countable

http://en.wikipedia.org/wiki/Uncountable_set

http://en.wikipedia.org/wiki/Cardinal_number

Noclevername

2008-May-01, 08:41 PM

But if it is possible ( and the universe is infinite ) then it DOES happen ...

Alternatively

if the universe is NOT infinite but it is possible then eventually it will happen.

The idea that "Infinite Universe" = "Events Repeated" has never, ever been proven. There can be nonrepeating infinities, too.

BioSci

2008-May-01, 10:54 PM

Something has to happen. We can easily construct a model where every individual possibility has zero probability, eg, pick a number on the real line, by spinning a pin, but one of them happens. All right the real world is imperfectly modelled by euclidean space. But if we are going to use probability as a model, then we have to acknowledge that in any model of continuous probability, things of zero probability do happen.

But this runs into a problem with definitions. It is very tricky to try and use infinities in probabilities since infinity is not considered a member of the set of real numbers i.e. 1/infinity is considered undefined - not zero.

Your example may sound like it is possible but there is no way to randomly pick one number from an infinite set of reals (go ahead and try...you will need to generate infinite precision to specify your randomly selected real number). The probability of selecting any one number is not zero - it is undefined.

Ivan Viehoff

2008-May-02, 08:30 AM

The notation "-->" stands for "approaches."

As n-->infinity, 1/n-->0.

As n-->infinity, n/1-->infinity

As n-->infinity, n/n remains equal to 1.

Thus, if the chance of something occurring is 1/n, and n=infinity, given an infinate universe, the probability of that event occurring is 100%, as n/n is 1.

You have to be careful with this, though....

You do indeed have to be careful, and you haven't been. Your sentence starting "Thus..." is not a valid probabilistic argument.

If the probability of something happens is 1/n, and there are m opportunities for it to happen, the probability of it happening is NOT m/n, even if the opportunities are independent of each other. For example, if we roll a fair dice twice, the probability of getting a six at least once are not 1/6 + 1/6, rather it is 11/36, as any basic text will show you. If the first argument were correct, then if we rolled a fair dice 7 times, the probability of getting a 6 at least once would now have risen to 7/6, which is nonsense. In fact the probability is 1-(5/6)^7. Though you might take comfort from that, since that tends to 1 too as the number of repetitions tends to infinity.

Edit 1: that last sentence wasn't quite right, actually less comfort for you. Consider 1-((n-1)/n)^n , which is the probability of at least one occurence in n independent attempts of something which has prob 1/n. That actually tends not 1 as n tends to infinity, but to about 0.632. There's something to think about. I suspect that this number is some combination of fundamental constants, but I can't guess what just now. But I'm equally sure that this is not relevant to anything. End Edit 1.

But in the universe, the opportunities are not independent: once something has happened it typically rules out, or alters the probability of, a lot things that previously might have happened. Once that atom has decayed, states of the universe in which that atom has not yet decayed are no longer accessible. The universe must proceed with a kind of continuity, it can't jump discontinuously to another state just because it is on our list of possible states.

All of you (frog march, tommac) who are making this kind of argument are making the same mistake. You are thinking that an infinite set must include everything. But one infinite set can be very sparse in another infinite set that contains it.

Let's consider time in small discrete steps. Then the process of time is like the counting numbers, a countably infinite set. You are thinking that if you have an infinite number steps, then it must visit every possible state of the universe. (Cardinality arguments having gone over your head). Now if the number of possible states of the universe were a finite set, then that could be true. But every possible state of the universe is itself an infinite set, and one infinite set does not have to visit every possibility in another infinite set. Indeed one infinite set can be very sparse in another infinite set, and I have already given several examples.

Edit 2: clear demonstration of the above. Consider the theory of random walks. http://en.wikipedia.org/wiki/Random_walk

In a one-dimensional random walk, we almost surely visit every possible location an infinite number of times, as we prolong the walk to infinity. That is the kind of argument you are looking for. But as the number of possible locations to visit increases, this argument breaks down. In a 3-d random walk, you are no longer guaranteed to get everywhere in infinite time: there are, in effect, too many possible places to visit, or more likely they are too dispersed, even though the order of infinity of possibilities is just the same as in the 1-d random walk. So you see, infinity need not be long enough to visit everywhere if there are "too many" or "excessively dispersed" places to visit. Clearly, the number of possible states of the universe is "more" and "more dispersed" than the mere number of possibilities in a 3-d graph. But the passage of time is precisely a kind of random walk through them one at a time. Such a random walk will not visit all possible locations. End Edit 2.

Another demonstration. Consider the situation where the energy of the universe, or some other state variable, tends assymptotically to zero over time. Thus the universe is infinite, and its energy gets lower and lower but never quite gets to zero. Then once the energy of the universe has fallen below a given level, then all the higher energy states have become inaccessible. Those high energy states were accessible for a time, but only a very small proportion of them were actually accessed during the finite time that they were accessible. Plainly, the states of the universe that happens, even if prolonged for infinite time, or in infinite space, is only a very small subset of the states of the universe that could have happened.

Or we could consider this in a very silly way by restricting what is possible down to a single state at any given moment. Ie, the only possible state of the universe at 0900 on 3 May 2008 is the one state that will in fact happen at 0900 on 3 May 2008. So every possible state actually happens...

Ivan Viehoff

2008-May-02, 11:08 AM

But this runs into a problem with definitions. It is very tricky to try and use infinities in probabilities since infinity is not considered a member of the set of real numbers i.e. 1/infinity is considered undefined - not zero.

Your example may sound like it is possible but there is no way to randomly pick one number from an infinite set of reals (go ahead and try...you will need to generate infinite precision to specify your randomly selected real number). The probability of selecting any one number is not zero - it is undefined.

It may be tricky but mathematicians have been able to do it for a long time now, and it is sufficiently well established theory that I studied the elements at high school before proceeding to university for my degree in mathematics. Its called continuous probability theory, or the theory of continuous random variables. It is the basis of statistics. It is the basis of stuff like the normal distribution, etc.

The example I give of spinning a pin to select a real number is well known. You put the pin with centre (0,1) then spin the pin, and choose the number by extending the line of the pin so that it intersects the x-axis. The number chosen will be distributed with the Cauchy distribution, which has a rather badly behaved pdf.

See

http://en.wikipedia.org/wiki/Continuous_random_variable

http://en.wikipedia.org/wiki/Probability_density_function

http://en.wikipedia.org/wiki/Normal_Distribution

http://en.wikipedia.org/wiki/Cauchy_distribution

Ivan Viehoff

2008-May-02, 12:25 PM

1/infinity is considered undefined - not zero.

I never mentioned or relied on 1/infinity. The closest I came was that I said that in a continuous probability density function, the probability of every point-like outcome is zero.

The limit as x tends to infinity of 1/x is always zero. So, although I never do, it is not totally unreasonable to appeal to such a concept of 1/infinity heuristically, since it can always be taken as zero for all practical purposes, (provided you haven't got any other infinities lying around). It is concepts such as infinity/infinity or zero times infinity that are utterly meaningless and can't be appealed to heuristically, since they can mean anything depending what you really meant.

Jeff Root

2008-May-02, 12:34 PM

Consider 1-((n-1)/n)^n , which is the probability of at least one

occurence in n independent attempts of something which has prob 1/n.

That actually tends not 1 as n tends to infinity, but to about 0.632.

HUH??????

I think you were right the first time.

-- Jeff, in Minneapolis

Disinfo Agent

2008-May-02, 12:41 PM

1/infinity is considered undefined - not zero.1/infinity is undefined in the real line, but it's zero in the extended real line. In higher mathematics, we work in the extended real line most of the time.

Your example may sound like it is possible but there is no way to randomly pick one number from an infinite set of reals [...]You're wrong.

I wish people would refrain from pontificating dogmatically on mathematics and probability until they've learned some of it.

Ivan Viehoff

2008-May-02, 12:44 PM

HUH??????

I think you were right the first time.

I wasn't. To convince yourself, I suggest you put the formula in a spreadsheet and try it. It converges nicely above about n=100,000. For very high numbers (over about 10^11) excel can't calculate it properly though.

Abelian Grape

2008-May-03, 04:59 AM

Just to throw a (virtual) monkey wrench into the discussion ...

Suppose that space-time is not represented by four real-number dimensions, as much of this thread has assumed. If quantum mechanics holds, is there not a minimum distance (Planck length) and time interval (Planck time)? In that case, particularly since it appears that the universe has existed for a finite time (<~14 thousand million years), the number of distinguishable space-time events, at least in the observable universe, is finite, albeit really large. And this will be true at any (finite) time in the future. This is quite apart from any hypothetical rip.

If the universe at large is actually spatially infinite, then the above does not hold; but this fact has no practical meaning to us.

Conclusion: for an event of sufficiently small probability, there are/will never be enough potential space-time events for it to occur within the universe we know. (What, never? No, never.)

BioSci

2008-May-03, 05:50 AM

1/infinity is undefined in the real line, but it's zero in the extended real line. In higher mathematics, we work in the extended real line most of the time.

You're wrong.

I wish people would refrain from pontificating dogmatically on mathematics and probability until they've learned some of it.

Yes, you have said this once before but you still have not given an example of such a "randomly" selected number from an infinite set. The question is not to demonstrate that some arbitrary number is a member of a given infinite set or the probability of data falling in a given range but rather to give an example of an algorithm that is capable of selecting any member of a given infinite set (essentially random).

Take a simple set of the real numbers from 1 to 2. Now generate a selection of "random" reals - do they have less than a billion digits each? Even such values would represent only a finite sample of the infinite number of reals that your algorithm needs to be able to select from... in fact any such algorithm would need to be able to generate number values with infinite precision - any finite limitation in the algorithm would mean that it would not be able to select from all possible members of the infinite set.

Do you think such an algorithm is possible?

Jeff Root

2008-May-03, 10:16 AM

Consider 1-((n-1)/n)^n , which is the probability of at least one

occurence in n independent attempts of something which has prob 1/n.

That actually tends not 1 as n tends to infinity, but to about 0.632.

HUH??????

I think you were right the first time.

I wasn't. To convince yourself, I suggest you put the formula in a

spreadsheet and try it. It converges nicely above about n=100,000.

For very high numbers (over about 10^11) excel can't calculate it

properly though.

My apologies. I read what I thougt you were saying instead of what

you actually wrote, even though what you wrote was completely clear.

I was thinking that the number of attempts was a separate variable.

In that case, of course, for a given n, as the number of attempts

increases, the probability approaches 1.

-- Jeff in Minneapolis

Disinfo Agent

2008-May-03, 01:24 PM

Yes, you have said this once before but you still have not given an example of such a "randomly" selected number from an infinite set. The question is not to demonstrate that some arbitrary number is a member of a given infinite set or the probability of data falling in a given range but rather to give an example of an algorithm that is capable of selecting any member of a given infinite set (essentially random).

Take a simple set of the real numbers from 1 to 2. Now generate a selection of "random" reals - do they have less than a billion digits each? Even such values would represent only a finite sample of the infinite number of reals that your algorithm needs to be able to select from... in fact any such algorithm would need to be able to generate number values with infinite precision - any finite limitation in the algorithm would mean that it would not be able to select from all possible members of the infinite set.

Do you think such an algorithm is possible?All your questions are answered in any solid textbook on probability theory. I recommend Feller, An Introduction to Probability Theory and Its Applications. It's a demanding book, but you're clearly a very clever guy, so that should pose no difficulties.

Ross PK81

2008-May-03, 01:42 PM

No, because impossible is impossible.

An infinate universe just means that absolutley anything that is possible will happen no matter how unlikely.

For example winning the lottery a milion times in a row. Or through a combination of faulty genes, a human is born which looks like an exact replica of the Empire State Building (on a smaller scale of course).

mugaliens

2008-May-03, 02:42 PM

Hmm. I won't comment on whether an event that does occurs can have zero probability; I'm rather unsure about measure of probabilities in an infinite domain. But I do wonder what, if any, mathematical abstraction is being to model infinite time: the real line, or some other construct?

I'd say that once we introduce real infinities as a hypothesis into the physical world, then we do get some tricky questions. One plausible idea is that everything not actually impossible happens (and infinitely often). Fair enough, but infinities tend to produce some very tricky logic (as has been pointed out).

Just one question: is the "time" in this infinite universe Archimedean, that is to say are every two instants in this infinite span a finite (but of course possibly very large) distance apart?

Now we've done it! We've left the real world and have begun talking about Metaphysics (http://en.wikipedia.org/wiki/Metaphysics)!

(it's a branch of philosophy dealing with the ultimate of everything)

You wrote:

I'd say that once we introduce real infinities as a hypothesis into the physical world, then we do get some tricky questions. One plausible idea is that everything not actually impossible happens (and infinitely often).

Wow. That's quite an excellent summary, and one that I agree with completely. Very nice work, agingjb!

mugaliens

2008-May-03, 02:46 PM

This being said ... the amount of energy in the universe may depleat.

Nope! It might be converted to matter, and matter may be converted to energy, both via E=Mc^2, but...

One outstanding question I still have involves matter and/or energy popping in and out of existance via quantum effects. To date (if I haven't merely missed it), I haven't seen a good answer to this, how much, whether it's precisely balanced, positive influx, or negative influx.

hhEb09'1

2008-May-03, 05:06 PM

Your example may sound like it is possible but there is no way to randomly pick one number from an infinite set of reals

You're wrong.

I wish people would refrain from pontificating dogmatically on mathematics and probability until they've learned some of it.I'm pretty sure that the ability to choose a random real number is more or less equivalent to the Axiom of Choice--after all, what you'd be doing is simultaneously choosing a single digit from each of a denumerable set of copies of the set{0,1,2,3,4,5,6,7,8,9}. OK, maybe not equivalent, but it would involve the Axiom of Choice, and mathematicians are aware that reasonable people can agree to disagree about whether we should assume that Axiom or not. Most people do, though, I think.

I would think that in a universe with truly infinite time, any event with a probability greater than zero must happen eventually.

No.

Take the example just above, choosing a random real number between 0 and 1, but instead of choosing the digits of our random real number all at once, we do it one step at a time, forever. If all real numbers are equally likely, then it is possible that the random real number that we'd be constructing will not have a single 4 in it anywhere. At each step, even after a billion non-fours, what is the probability of getting a four on the next step? Still only one tenth, just as at the first step. We perform the action an infinite number of times, but it never occurs.

Otherwise, that particular number would not be constructable this way--and we know that it is just as likely as any other number.

Disinfo Agent

2008-May-03, 05:08 PM

I'm pretty sure that the ability to choose a random real number is more or less equivalent to the Axiom of Choice--after all, what you'd be doing is simultaneously choosing a single digit from each of a denumerable set of copies of the set{0,1,2,3,4,5,6,7,8,9}. OK, maybe not equivalent, but it would involve the Axiom of Choice, and mathematicians are aware that reasonable people can agree to disagree about whether we should assume that Axiom or not. Most people do, though, I think.I have no idea if you're right, but that's interesting. I'd never thought of it that way.

P.S. Wait a minute, isn't the Axiom of Choice only a problem for non-denumerable choices? I seem to recall something of that sort...

P.P.S. There may also be something to say about the word "choice". If you build a random experiment, and then measure a random variable associated with it -- whose value you thus cannot predict in advance -- does it count as a choice, in the sense of the Axiom of Choice?

hhEb09'1

2008-May-03, 05:27 PM

P.S. Wait a minute, isn't the Axiom of Choice only a problem for non-denumerable choices? I seem to recall something of that sort...The Axiom of Countable Choice (http://en.wikipedia.org/wiki/Axiom_of_countable_choice) might come into play where you do not need to assume the complete Axiom of Choice. But, as it says at that wiki page, it is still an axiom, not provable in Zermelo-Fraenkel set theory.

P.P.S. There may also be something to say about the word "choice". If you build a random experiment, and then measure a random variable associated with it -- whose value you thus cannot predict in advance -- does it count as a choice, in the sense of the Axiom of Choice?Selection rules can work around the Axiom of Choice, but I'd doubt that a random selection rule would. :)

After all, what else is there? :)

Disinfo Agent

2008-May-03, 06:05 PM

In the article on the Axiom of Choice (http://en.wikipedia.org/wiki/Axiom_of_choice), though, they have the following:

Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box. Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. The number of boxes is finite, so eventually our choice procedure comes to an end. The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on. (A formal proof for all finite sets would use the principle of mathematical induction.)

For certain infinite sets X, it is also possible to avoid the axiom of choice. For example, suppose that the elements of X are sets of natural numbers. Every nonempty set of natural numbers has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set. This gives us a definite choice of an element from each set and we can write down an explicit expression that tells us what value our choice function takes. Any time it is possible to specify such an explicit choice, the axiom of choice is unnecessary.

hhEb09'1

2008-May-03, 06:18 PM

In the article on the Axiom of Choice (http://en.wikipedia.org/wiki/Axiom_of_choice), though, they have the following:

Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box. Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. The number of boxes is finite, so eventually our choice procedure comes to an end. The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on. (A formal proof for all finite sets would use the principle of mathematical induction.)That concerns only finite choices

For certain infinite sets X, it is also possible to avoid the axiom of choice. For example, suppose that the elements of X are sets of natural numbers. Every nonempty set of natural numbers has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set. This gives us a definite choice of an element from each set and we can write down an explicit expression that tells us what value our choice function takes. Any time it is possible to specify such an explicit choice, the axiom of choice is unnecessary.

Yes, that is what I was referring to in this comment:

Selection rules can work around the Axiom of Choice, but I'd doubt that a random selection rule would. :)

A random choice is not such an explicit choice. An explicit choice would not result in a random choice, always choosing the largest one for instance would result in .999999...

Aaaaiiiieeeeee!!!

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