Unlikely things in an infinite universe

Factual Questions
41

Sorry, I should have noted that I was referring only to radioactivity from elements on the ground.

Although, it’s also possible that somewhere out there is a planet that has never intercepted a cosmic ray — again, purely by chance.

Hrmmm? If a lone uranium atom can remain intact (fail to decay) over a 4 billion year time-span, then what’s to stop a trillion trillion of them from doing the same? Ignoring the outlandishly low probability of such a thing, that is. Do they mediate each other’s state, like ripening bananas?

And likewise, if an unstable atom can decay within a millisecond of its formation, what’s to stop all the others on the planet from doing the same? Again, aside from the phenomenal luck that would be required?

So I don’t agree that this would be a violation of any physical law. Radioactive decay is a probabilistic behavior, like the laws of thermodynamics. There is no law preventing a room full of comfortable air from self-segregating into two regions of scalding hot and freezing cold air on opposite sides of the room. It’s just extremely unlikely — so unlikely that you might as well call it effectively impossible.

But in this thread we are contemplating an infinite universe, right? Suddenly the odds of such bizarre events are looking up.

42

Er, I retract the previous statement. The likelyhood of any measurable shift in the statistically predicted half-life of a significant sample is virtually infinitesimal, but since we’re talking about spontaneously arising 747’s and organically evolved VW Bugs I suppose the laws of statistical mechanics are right out the window here.

Still trying to figure out how to make the hostess’ undergarments jump one foot to the left, though.

Stranger

43

Actually, I’m pretty sure the same is true for an infinite list of random integers, even though there are “less” of them. As long as the person writing the list is choosing at random from all integers, not just the ones between, say, -1 million and +1 million, every given integer has a zero probability of showing up next. I think.

44

Not every integer has probability zero of being picked; furthermore, they don’t all have the same probability.

45

What if I insist on a full size spare tire?

46

Yep. We’ve got quite a few on this planet as a matter of fact. Convenient, eh? Sure, it took a few billion years, but we’ve got 'em as of a few decades back.

You know - random galaxy, random planet, random evolution, random exploration of the New World, and some random engineering decisions… :cool:

We’re talking infinity. Infinity is reaaaaaally big! Seriously.

Here’s a classic hypothetical example involving a smaller “universe”. Namely, your computer monitor. Suppose we write a screen-saver that randomly keeps changing the pixels. Now let it run for an infinite amount of time and assign an observer to watch for all eternity. All possible combinations of pixels will eventually appear in no particular order or pattern.

or in other words you will see:

  • A photo of every location on the Earth, from every possible angle, at every possible time. Inside bedrooms, from the Moon, sunrise on March 23rd 102 AD in what is now downtown New York, the actual crucifixion of Jesus, a dinosaur fight from 100 million years ago, your own birth.

  • Screenshots of every frame, of every game/spreadsheet/text editor/OS/whatever that ever existed or will exist. Pages from all novels ever written, lines of source code from every program. Every webpage in its every iteration.

  • What all species of everything looks like, every insect, plant, animal, alien, whatever. With diagrams and explanatory notes.

  • Pictures of all the StraightDope members, (actually all inhabitants of the Earth) for every second of their lives, with their thoughts superimposed on the screen in text. In every possible language.

-…

You get the idea. However, you also will see a unbelievable metric crapton of kaleidoscope-like static and un-interpretable images. Such is the nature of entropy, probability, and infinity.

You can cheat a little bit by programming the screensaver to cycle through the combinations methodically instead of randomly; this changes the time it takes from infinity to “countable infinity”.

47

That is just a different way of stating the infinite monkeys at typewriters with infinite time eventually coming up with War and Peace (though they might come up with the screenplay to Die Hard significantly sooner).

Unfortunately the things that happen in the universe are not completely random events. I’m not talking about the Hand-O-G*d™ but the things we all agree can be quantified. I mentioned before the significantly less than random odds of a bunch of atoms bumping together to form a 747 because many forces would prevent that from happening. A forged metal part or even a bit of sheet aluminum isn’t just a bunch of atoms that happen to be stuck together. The process to make them rearranges the crystal structure which give the parts a lot of their stretngh. Okay, that is a bit nit picky I admit. It sounds like finding a flower that grows a perfectly formed 1953 VW beetle but complaining that they left out the bud vase but it is a legitimate argument against a bunch of atoms just randomly forming together to form an object.

But given a different tack what if someone else evolved to learn the metalworking arts, could they also build a 747 exactly as we did? Unlikely unless the parameters they start with are the same. Does their planet have the same amount of gravity? Does their atmosphere have the same density? Those factors have a huge impact on how an airplane design comes out. A species that evolves under different conditions would invent airplanes that may not even work on this planet.

48

In quantum mechanics, it is generally the case that any physically-possible outcome can occur with non-zero probability. If it is possible in principle that something could happen through a massive conspiracy of cosmic coincidences, then in an infinite universe the chance that it will never happen anywhere is zero.

49

keep in mind that most of the things discussed in this thread are never going to happen, not because of the odds against them but because of the odds of them happening in the time remaining to the universe…I think that made sense.

basicaly given an infinite number of people searching for a “naturally occuring 747” you wouldnt find one.

50

I think people are underestimating infinity. Axiom: If the probability of an occurence is finite, however small, it must occur given infinite chances to reach it. Or as Jasonfin gives in the more formal statement: “the probability of it never occuring anywhere is 0”

In general, the mechanism of star formation from gravity and fusion, random gas swirling around etc would not produce 747s directly of course. You’re right, you can’t stack the deck enough to make it happen as the simple nature of the process can’t spontaneously do it.

But it can make planets, which can evolve life, which can build them. The chance of this occurring is non-zero for sure since it has occurred here. Given infinite chances, it’ll happen again for sure as well.

51

I’m not so sure this is true. Something can be possible and still have a probability of zero (in the sense of your “axiom”) so that there’s no guarantee it will happen.

That was the point of my earlier remark:

Give me any infinite list of real numbers. Now look at any one particular number on that list. Obviously, it’s possible for that number to appear, because there it is. But there’s still 0 probability for that particular number ever to occur on such a list.

52

So let’s talk for a minute about probability 0.

A probability is function p [symbol]Î[/symbol] 2[sup]E[/sup] [symbol]®[/symbol] [0, 1] for some set E which satisfies the following three conditions:

  1. p(E) = 1.

  2. p([symbol]Æ[/symbol]) = 0.

  3. For any list of sets A[sub]0[/sub], A[sub]1[/sub], A[sub]2[/sub], … with A[sub]i[/sub] [symbol]Í[/symbol] E and A[sub]i[/sub] [symbol]Ç[/symbol] A[sub]j[/sub] = [symbol]Æ[/symbol] iff i [symbol]¹[/symbol] j, p([symbol]È[/symbol]A[sub]i[/sub]) = [symbol]S[/symbol]p(A[sub]i[/sub]).

As an aside, this definition gives us something that satisfies the Kolmogorov axioms. Those are the standard way of describing probabilities, but I think this is a little easier to understand.

The notion of p as a function should be pretty intuitive, and I think conditions 1 and 2 are trivial to understand. Condition 3 is interesting here. What that basically says is that if you have a countable number of pairwise disjoint sets, you can just add up their probabilities to get the probability of their union. What this implies is that for a finite or countably infinite set E, if you compute [symbol]S[/symbol]p(A[sub]i[/sub]) with A[sub]i[/sub] [symbol]Í[/symbol] E and the restriction that p(A[sub]i[/sub]) [symbol]¹[/symbol] 0, it comes out to be 1. That’s where you get the notion that events of probability 0 are impossible–they just don’t contribute anything to the probability of another event.

Now, what if E is uncountably infinite? Take the set [0, 1], and define p(S) by [symbol]ò[/symbol][sub]S[/sub]dx. I’ll leave it as an exercise to the reader to show that p satisfies the conditions above. The question of interest here is what is p({h}) for h [symbol]Î[/symbol] [0, 1]? Well, {h} = [h, h], and p([h, h]) is [symbol]ò[/symbol][sub]h[/sub][sup]h[/sup]dx, which is clearly 0. Therefore, in an uncountably infinite sample space, discarding all the events of probability 0 may leave you with no events.

An infinite universe may be viewed as an infinite sequence of events, and even if there are only a finite number of possible events, you get an uncountably infinite number of possible sequences. Therefore, you can’t just dismiss events of probability 0 as impossible.

Here’s the take-home lesson: Given infinite time, every event that has a non-zero probability of occurring occurs with probability 1. That does not mean that it is guaranteed to occur.

53

Before us, there was no life here that we know of. Yet we have 747s, so it must be possible.

54

Feel free to correct me on this, but AFAIK this is the difference between finite probability and infinite probability.

There are an infinite number of reals between 0 and 1. Pick one at a time, never pick the same one twice, and do it infinitely. You’ll never pick them all, because there are an infinite number to choose from. So you are absolutely right, a specific number may never come up - I’m not sure it’s possible to say how probable it is that a particular one will be picked. 0 is the best answer.

Something like flipping a coin, or the changing pixels in my computer monitor example is finite. Every state will be reached because there are a finite number of states, and with an infinite number of tries; effectively one never stops trying while also picking infinitely fast.

Does the universe have finite states?

I’m not sure, but from what little I know about quantum mechanics, I somewhat suspect that all measurable quantities are at some level granular. Energy certainly is (planck constant), and if matter is condensed energy from which we get our basic forces and effects on space then it seems plausible…

Anyway, if it is finite than it must be possible to go through all configurations in infinite time, even to the appearance of a “747 nebula” in deep space, just as RevTims’s coin will eventually come up heads. The difference is then just a matter of degree.

I agree with Ultrafilter that if the sequence of states is important, than infinite time gives infinite possible sequences again.

55

That’s a red herring. The point is that the probability of picking the first number that you got was 0–and I am absolutely certain that this is the right answer.

This is not true, as I demonstrated above.

56

If it’s random, then the time needed is countable infinity. If it’s methodical, then the time needed isn’t infinite at all. For my computer monitor, for instance, it would take a mere 16777216[sup]1310720[/sup] iterations to complete, a number which is far smaller than the vast majority of all integers.

As for some earlier objections, if something doesn’t occur at all in an infinite universe, how does one justify saying it’s possible? Define “possible”.