Yes, relative frequencies. It wouldn’t have made much sense to say “some common events will happen a really big infinite number of times and some rare events will only occur a small infinite number of times.” I felt that expressing it in billions and thousands got the point across better.
Or here’s another way of putting it. If Pi is an infinitely long sequence, is it inevitable that at some point A will appear in the sequence? The answer is obviously no, Pi is infinitely long but it’s composed entirely of only ten digits. Infinity composed of a finite set of possibilities.
I got ninjaed to the good, funny answer so I’m stuck with the boring one.
The point of the metaphor to digits in an expansion is the presumption that all events can be encoded in that way. The expression of that code doesn’t matter. Pi can just as easily be written in binary, leaving out 2 through 9. But all of Shakespeare can be expressed as a binary code, which of course computers already do. And a movie of a Shakespeare play can be expressed as a binary code, because that’s what a DVD is. And the reality of a troupe of players acting out Shakespeare can be expressed as a binary code, because we can put every one of their atoms over time into a series of states that would replicate the performance if magically rewound.
We haven’t yet proved that pi is normal, but that’s another so what. Most random decimal fractions between 0 and 1, that is, an infinite number of them, are going to be normal. If you select a fraction at random you will get one of these. The same logic applies if the multiverses are truly infinite. If you have an infinite number of random universes, you should presume the equivalent of normality. In that case every finite number string, however it is expressed, will appear, and will appear an infinite number of times. You can’t get around it. Infinite means forever, never ending. There are no relative frequencies in an infinite number line. Every possible countable thing happens an infinite number of times, the same level of infinity. In infinity everything is equal. That’s why I can predict that all of these events, however unlikely in our finite universe, will happen infinitely often in infinity.
IOW, in our universe, the odds that the states to create a 1981 Pontiac Trans Am will suddenly happen through quantum fluctuations are too small for any practical purpose. But those states will appear an infinite number of times in infinity. The math guarantees it.
Our brains really aren’t set up to think that way. That’s probably why poor Cantor went mad after a lifetime of grappling with infinity - or maybe madness explains why he was able to crack the problem in the first place. For over 100 years people have been assaulting infinite math because they don’t like the inevitable implications. And nobody’s ever been successful. The math is beautiful and it works and it tells us things that are beyond our imagination.
*To real mathematicians: I’m aware I’m using random and normal loosely as part of the metaphor. Feel free to add rigor and stir.
If you keep all the parameters of your bag a constant, and you keep shaking it for a truly infinite amount of time, eventually (after an inconceivably long time), you really will get your flamin’ chicken Pontiac (as well as a very large number of near-misses, like one where your chicken is chrome instead of gold, or one with the steering wheel on the right side, or one that’s missing three of its hubcaps, or a VW bug with the leather seats etc.). The catch is that, in our real Universe, you wouldn’t be keeping the parameters of the bag constant. Obviously, you’re more likely to get your Pontiac when the material in the bag is starting off at close to Pontiac-density, and less likely as the density gets further away from that. But in the bag that is the Universe, the density would be continually decreasing, so every moment that you don’t make your Pontiac, it gets harder to pull it off on the next try. And meanwhile, you’re also shaking the bag less and less hard as time goes on.
I found the question interesting. I want to raise a specific version of it. Let’s assume that in the fullness of time, all the black holes have evaporated, all the hadrons have decayed into the lightest one, the universe has reached its entropy death. For each volume of space, and each particle, there is a certain probability that it is found in that volume. Putting these all together, there is a certain small, but not quite zero, probability that it is in exactly the same place and state that it is at this very instant. Now. Will this happen given an infinite time? Will it happen infinitely often? This seems like a really interesting question.
Well, it depends on whether the Universe continues to expand until all things are dispersed and peter out to the point of nothingness before the big bang OR the universe goes through infinite cycles of big bang-expansion-contraction. If the latter is the case, theoretically “I” or “the Pontiac Trans Am” should happen again at a certain point in time, regardless of how long it may take (if any perception of time even matters in these terms), no?
I was taking “infinite time” to implicitly imply a perpetually-expanding universe. Truly cyclical universes are more a topic for after-hours BSing than actual models.
**How **do you choose a random real number? I’m happy to be proven wrong, but I’m not convinced that’s a meaningful action. Or at least you will never have the opportunity to choose a second one because it will take you forever to choose the first one.
This is a mathematical operation, not a real world one. Choosing a random irrational number between 0 and 1 is well-defined. It’s being used correctly.
Consider the square [0, 1] x [0, 1] (i.e., a square of side-length 1).
For every positive integer n, there is a way to split the square into n^2 many equal-sized squares, of which only n are needed to cover the entire line from (0, 0) to (1, 1). Thus, that line has area no greater than n/n^2 = 1/n, for each positive integer n (i.e., the diagonal line between opposite corners has infinitesimal area).
The relevance or lack thereof of this mathematical claim to whatever issues you are interested in exploring I now leave to you.
(Eh, being very nitpicky, I feel I should have phrased the above with [0, 1) instead of [0, 1] (and similarly for the smaller squares, so that they have no overlap). But whatever…)
Except that it isn’t “inconceivably low odds”, it’s “infinitely low odds”. Consider an extremely simple “object” defined by two Helium atoms that are X distance appart. It is equally likely that there is another similar such “object” where the two atoms are Y apart, where Y is not equal to X. Also equally likely is any “object” where the distance is any number between X and Y. But that defines an infinite number of “objects”. How do you suppose that works out for an “object” as complex as the entire Earth?
I think you’re right, but I’ll make a devil’s advocate case for multiverses that feature “clustering,” so that “all possibilities” fail to arise. For instance, universes within a multiverse might only arise as “daughter” universes, differing from the parent universe in the results of some quantum measurement or event. e.g., an atom splits, deep in the rocks of a mountain in Borneo, and the event might emit an “up” neutrino or a “down” neutrino. (Or whatever…) The daughter universe still resembles the parent universe to nearly the degree of identity. Obama is still president in each; The Straight Dope is still fighting ignorance; the Chargers still suck. This high degree of “genetic” resemblance leads to clustering, and thus to limited possibilities of universes where Beyonce is president, The Straight Dope is a drugs-and-porn site, and the Chargers are in the Super Bowl.
That does not mean that the square [0, 1] x [0, 1] is very much smaller than the line segment [0,1] or any smaller at all. Both of those items are infinite sets and we can tell if they are the same size (same cardinality) only by putting them into a one-to-one correspondence with each other. Just because the line can be put into a one-to-one correspondence with a subset of the square doesn’t mean it has a smaller cardinality. After all, all square integers can be put into a one-to-one correspondence with a subset of all integers (obviously with themselves), but they can still be put into a one-to-one correspondence with all integers n[sup]2[/sup] maps to n. And in fact the set of points in a square can be put into a one-to-one correspondence with one of its sides. See, e.g., Theorem 18 in http://www.earlham.edu/~peters/writing/infapp.htm
I am aware of all that. Cardinality is not the only notion of size, and certainly not always the relevant notion of size. (There’s nothing wrong with saying “A line inch long is much, much smaller than a line five billion miles long”, even if they have the same cardinality…)
[Not that I made any claims about the relevance of the area/uniform probability measure I noted either.]
[ol]
[li]That a duplicate configuration of a pair of helium atoms exists is likely.[/li][li]That a duplicate configuration of a million helium atoms exists is less likely[/li][li]That a duplicate configuration of the Earth exists is less likely still[/li][/ol]
But in an infinite universe, this plays out as there being (infinite) cases of all three, but proportionally more cases of 2 than 3, and proportionally more cases of 1 than 2.
[ol]
[li]That a duplicate configuration of a pair of helium atoms exists is likely.[/ol][/li][/quote]
The number of configurations for two Helium atoms separated by a distance is infinite, not finite. Therefore the probability of duplicating even a single instance of two Helium atoms at a specified distance is zero.
It’s a metaphor so don’t strain it too far. I didn’t intend it as a mathematical statement. My point was that a small finite set of events can occur over and over again for an infinite length of time. So infinite time does not necessarily mean infinite variety.
To strain your metaphor, binary code could also theoretically express a troupe of players performing Cats. And, imitating the Broadway run, it could go on forever. Literally in this case. The performance could be repeated over and over again for infinity. Unlikely? Yes, but as you pointed out not mathematically impossible. And in this infinity, there will never be a performance of Shakespeare. So this proves that an infinite amount of time does not guarantee everything will eventually happen.
This is true only if space is not quantized. That’s why I specified Planck length and Planck time above. Once you have a minimum physical distance and time, then only a finite number of states are possible, as in my earlier calculation.
This could happen, in the sense that 1/7 has an infinite but repeating decimal expansion. My point, again, is that a random assortment of universes in infinity can be assumed to be like the relationship between the rationals and reals. There are so many more reals between 0 and 1 than rationals that the likelihood of picking a rational randomly is 0. Once you start playing around with an infinity of universes you have to accept that all possible states will occur each an infinite number of times. You can’t get out of it. Your proof isn’t a proof.
