It does indeed, thanks.
-FrL-
Well, I don’t know. Also keep in mind that while mathematics are used to model the physical world, they really exist in an ideal world, so they are not applicable without some approximations. Probability theory is useful, but it’s worth remembering that it is an abstract theory, and I’m not convinced that it can be used to gain information about the physical world without some loss. And I must admit that the concept of an infinite universe is quite difficult to apprehend for me.
You’re welcome.
As I said, it might happen once. But have it continuously and predictably happen? It takes more than chance, I would think.
And now that you have specified that you are talking about THIS universe, I don’t think our universe is infinite in the way to need to to be. I am going to need someone who knows what he is talking about to back me up here, but I believe that, although it occupies an infinite (or is it boundless?) space, there is not an infinite number of particles in it.
And as others have explained, no matter how many infinite times you throw an infinite number of dice, you cannot ever get a double 7.
Someone above glissandoed over one of the problems with the phrase “infinite universe” and no one noticed.
“The universe is approximately 11 billion years old.”
That is old, but it ain’t infinite. Even a “multiple bang” universe, with events separated by more than 11 billion light years, (or more than 13, or 34, or whatever your favorite age of the universe is) really old is not infinitely old. So, any configuration of matter that requires more than the age of the universe to occur, has not yet occurred. Not only that, but to exist, it must have had a precursor configuration which the uniform laws of physics cause to become the new configuration. Every configuration is a resultant of a prior configuration. Configuration that cannot arise from prior configurations cannot exist, and the size does not alter that.
Tris
Its entirely possible though extremely improbable for there to be a region of space in which all conscious agents, upon willing themselves to levitate, do in fact levitate. I mean to be describing a situation in which there is in fact no actual lawlike connection between their willing themselves to levitate and the actual levitation. Its just a huge and amazing coincidence. (Though the conscious agents themselves might think incorrectly there’s a physical law at work.) Such a situation is very, very improbable in the extreme, but not, for all that, impossible.
Think of it this way: Its just barely possible, though to a ridiculously small probability, that there’s no such thing as gravity, and that all the “gravitylike” interactions we’ve observed have simply been a huge and amazing coincidence. There’s nothing about the fact that things have acted “gravitylike” in the past to force a conclusion they will continue to act gravitylike in the future, and even more to the point, there’s nothing about their having so acted in the past to force a conclusion that they are in fact doing so because of any physical law.
The conclusion is a safe one–indeed, its the right one to make–but it is not an absolutely necessarily true one.
-FrL-
Can you give me an example of what you mean by this? What sort of configuration is impossible given the current age of the universe?
(BTW, we have already observed distant objects that are farther apart than the distance light has travelled since the birth of the universe.)
After some thinking, I’m going to throw out the hypothesis that in order to cycle through all possibilities you would need a moving system. In an infinite system, there is no loss or gain (after all, you can’t add more infinity), so there is no energy propelling things.
Eventually you’ll hit a permutation that is self-sustaining, and no further permutations will be moved into. So long as at least one permutation of all possible configurations is a self-sustaining system, and there are no ways to add or subtract mass/energy from the universe, you’ll end up in an unbreakable set state.
I hadn’t heard that. How can I follow up on this?
-FrL-
Tens of thousands of quasars have been found in all directions from Earth. Most of them are between 3 billion and 12 billion light years away. Current consensus puts the age of the universe at about 13.7 billion years.
Look in one direction and find a very distant quasar. Look in the opposite direction and find a second very distant quasar. If you add their distances together, you’ll probably get something larger than 13.7 billion light years. We can see them both, but they can’t see each other. The light hasn’t had time to get there yet.
(And, in fact, they may NEVER see each other. Since the universe is expanding they’re already much further away than they appear to be. And if the rate of expansion is increasing due to dark energy, the light may never get there … .)
Does this mean the two quasars do not come from the “same big bang” so to speak?
-FrL-
No, they were part of the same big bang. They just moved away from each other at faster-than-light speeds during the inflationary epoch.
What is “countably infinite”? Is it possible to count infinity?
As a side note that is almost, but not quite a hijack:
A pet peeve of mine is people referring to multiple universes or multiverses or any pluralization of “universe” whatsoever. This is only an uniformed opinion of mine as I know next to nothing about physics and cosmology, but it seems to me most helpful to think of “universe” as “all that there is, was, or will be” in other words, all of spacetime.
If there are multiple universes then what do you call the collection of all of them? “Cosmos” maybe? To me “cosmos” is the same as “universe” is the same as “everything”.
A “countable infinity” is an infinite set that can be put in one-to-one correspondence with the integers. Higher-order infinite sets (like the set of all real numbers, or the set of all curves that can be drawn in a plane) don’t have this property.
If a set S is countably infinite, there is a function f from S to N (i.e., {0, 1, 2, … }) with two properties:
[ol][li]f(x) = f(y) implies x = y.[/li][li]For every n in N, there is an s in S with f(s) = n.[/ol][/li]See here for a demonstration that not every infinite set is countable.
I can argue with you about this all day. The probability is 0 and that does mean it’s impossible. It’s like saying it’s possible for a rock to be exactly 1 centimeter wide. It may be pretty damn close, but unless you specifically defined the centimeter based on the width of that rock, then you will measure it at something other than 1 centimeter if you have enough precision. There’s no telling how much precision will be required, and it may be beyond our technology to measure, but it’s not exactly 1 centimeter wide. That’s absolutely certain. It’s also absolutely certain that if you had a random letter generator that could go on for infinite time, even if there’s only a one in a googleplex chance of C showing up each time, that you would not only have at least one C, you would have infinity of them. There’s no telling when the first one will show up - it could be after humanity has died off - but it will show up, and then another, and another…
Sorry, but you’re wrong. In an uncountably large sample space, the probability of a possible event may be 0.
Like any other measure, a probability function P is countably additive. What this means is that if you have a set I, and you break into a finite or countably infinite number of pieces I[sub]j[/sub] such that the intersection of any two of them is empty, P(I) = P(I[sub]1[/sub]) + P(I[sub]2[/sub]) + … There is no similar requirement that a probability function be uncountably additive, and that’s a very good thing, as you’ll see in a minute.
The upshot of this is that if you have a countable or finite number of possible events, then P( { E | P(E) = 0 } = 0. If you have an uncountable number of possible events, then that equation is no longer true. In fact, for the set of infinitely long strings over the alphabet {A, B, C}, the probability of every single outcome is zero.
The lack of uncountable additivity is a very good thing. If measures were uncountably additive, then the length of the interval [0, 1] would be zero, as it’s the union of uncountably many single points, all of which have length 0.
Oh sure, if you’re in the physical world, which, as I’ve said, can never be perfectly modelled with mathematics. In the ideal mathematical world, you can have a cube that’s exactly 1 cm wide, and you can have valid events that have probability 0 of occurring. That’s why we say that events with probability 1 will occur “almost surely”.
Well, the thing is: is it meaningful to say that you have a random letter generator working for an infinite time in the physical world? I’m not sure of that. So we have to go with an idealized situation. And in the ideal world of mathematics, there is a probability 1 (i.e. it is almost sure) that you will find at least one C (or, indeed, infinitely many Cs) in your sequence, but that doesn’t change the fact that a sequence containing only As and Bs is a valid output of your random letter generator.
I understand what you’re saying. I’m just pointing out that in mathematics, events that have probability 0 of occurring are not necessarily impossible events. Of course, in practice, they are negligible.
But that’s just… wrong. I don’t even know why you would say it. “Absolutely certain” means “It is not possible for it to be false.” But it is possible for there to be a rock exactly one centimeter wide*, and it is possible that this rock be measured.
I have already explained why this is wrong, without recourse to talk of probability. If each individual selection can select a non-C letter, then it is possible that, no matter how far down the list you go, you never come up with a C.
-FrL-
*This assumes matter actually has exact length, which is probably not correct, but that’s not the point you were getting at
One: An example might include a large region of space where half of the protons have decayed. Since we are still searching for an example of one proton decaying withing our horizon, it seems unlikely that such a region can exist yet.
Two: The fact that two objects appear to be in opposite directions from us, and are 13 billion light years from us, does not actually mean that they must be 26 billion light years from each other. Cosmic geometry is not quite that simple. (Nor do I claim to be able to explain it.)
Tris
But protons, if they do decay, are expected to decay into familiar particles. A region of space where half the protons decayed would be indistinguishable from a region of space that had half the number of protons to start with, plus a random scattering of other particles.
In any case, radioactive decay is a probabilistic event. A proton may have a half-life of 10^35 years, but in an infinite universe you would expect to find regions where purely by chance large numbers of protons had decayed much more quickly.
Within the observable universe space is largely flat. True, it may be warped slightly by mass concentrations but those effects are trivial on the cosmic scale. If we observe one quasar that appears to be 10 billion light-years from us in one direction, and another that appears to be 10 billion light-years from us in the opposite direction then, at the time the light from both quasars started on its journey, they really were 20 billion light-years apart.
Of course during the 10 billion years it took the light to reach us both quasars continued to recede. Although they appear to be only 10 billion light-years away, they’re actually more like 15 or 16 billion light-years away now, putting them more than 30 billion light-years apart from each other. If the expansion of the universe accelerates they may never see each other.