Infinite universe - Jumping to pointless conclusions I watched an episode of thee BBC Horizon series titled 'To infinity and beyond'.
In this program a number of respected physicists and mathematicians were talking about the nature of infinity and an infinite universe.
They argued that if the universe is infact infinite then every possibility must play out somewhere, even going so far as to get a physicist to calculate how far you would have to travel to find an exact replica of the earth.
They also used the classic example of infinite monkeys producing the works of Shakespeare.
Now to me the monkey example makes sense considering that the works of Shakespeare are finite and so are the number of possibile outcomes of a monkey mashing away at a keyboard for a given period of time (although very large still finite).
But the logic of applying this to the universe seems flawed to me, surely in the context of the universe the possibilities are also infinite as it is with a sequence of numbers.
the numbers 1,2,3,4... go on infinitely without repeating themselves so to say because something is infinite every possibility must play out seems to me incorrect.
Now perhaps you could argue that a finite number of atoms can only assemble themselves in a finite number of ways - again I can see this although I am not entirely sure that two lots of atoms arranged in the same way are necessarily the same since we currently rely on probablity calculations to predict the outcome of subatomic processes (this could arguably be due to slight variations at a more fundamental level...) but even if this were true. They went further to suggest that in an infinite universe there is an exact copy of you or I doing exactly the same thing and indeed every possible thing. 
Now it seems to me at this point things have become ridiculous, even by the time you have calculated the possibility of an exact atomic copy, the number of possible interactions I could have at any time with the universe, their affects on my atomic makeup and indeed future interactions, surely at this point we are dealing with infinite possibilites.
Any feedback on any of these points welcome but my central assumption here is this: Infinite possibilities will not play out even over an infinite period of time. 
Is this correct?
 A: You're quite correct that assuming space is continuous there is an infinite way of arranging the atoms that make up e.g. me. But there are two reasons why a copy of me isn't infinitely unlikely. Firstly on a practical level, if you moved around some of the atoms in me no-one would notice, so you don't have to match me exactly. That means somewhere in an infinite universe there is an approximate copy of me, or indeed an infinite number of approximate copies of me.
On a more fundamental level, while most physicists believe that space is continuous in the sense that it's not a lattice, you can't define positions of atoms to infinite precision because you (probably) can't define them more precisely than a Planck length. So even if you're not happy with an approximate copy of me, there is a finite probability of finding one that is physically indistinguishable from the original. Again there are actually an infinite number of such copies.
All this is straightforward mathematics, but the assumption that the mathematics is realised in nature obviously cannot be tested, so it remains the sort of thing that's fun to talk about towards the end of an evening in the pub, but nothing more.
A: WebweaverD, good question and good discussion already. However, I think you already answered your own question: As shown by your own example of infinite and non-repeating integers, the question of "infinite exploration" all depends on how you first set up your rules of expansion. It is trivial to come up with rule sets that expand forever, yet can be proved formally to explore only a very narrow range of local configurations.
However, there are other remarkably simple rule sets that while repetitious in some ways -- look at the start and ends of the link I just gave -- are always exploring slightly different variations of earlier themes. Mandelbrot is a literally gorgeous example of this kind of recursively fractal symmetry breaking, where things can look so similar that it is almost impossible to tell them apart... and yet they are not. The deltas, while tiny, become incredibly significant over time. Such initially simple rule sets also sometimes land in places in ways that are not easily predicted, a point that the video makes by showing strikingly unexpected regions as well as familiar looking ones.
And lastly, variant rules can sometimes sort of explode into curious explorations of what amount to mini-fractals with their own unique properties, the fractal equivalent of a baby universe.
I do not know if anyone has devised encompassing proofs of when and whether a particular rule set will diversify in truly interesting ways. I know a physics expert in chaos and fractals and will try to remember to ask him about it next week. Also, cellular automata experts like Andrew Ilachinski (he wrote an excellent mathematical textbook on the topic) might have ideas about limits.
Now, how does all of this relate to physics? Well, how complex is the Standard Model that captures all of known physics except gravity? Does it have enough diversity to ensure that any exploration of related possibilities could, at the very least, demonstrate the vast richness of the Mandelbrot set?
Well, sure. And if you factor in the chaos in our universe of how its rules are applied, we seem to be in a pretty pattern-rich situation. I would assume but have not looked it up that the exploration-intensive branches of physics interpretation would say that is because we live in one of those "explosions" I mentioned, a region where everything came out just right.
My fascination with the Standard Model is simpler, however: It really does look like a structure that has what I would call recursive symmetries, that is, of rules that are applied fractal-style to earlier results until the result is something beautiful, rich, and capable of generating extreme diversity. This is one reason why I'm not personally interested much in either string theory or quantum gravity. Those to me look too much like deeper recursive elaborations of a rule set that is already in full play in the universe we know best from a combination of quantum physics and astrophysics.
So: Maybe it's not an infinite exploration, but I do think there's a good chance that somehow, somewhere, our universe glommed onto a set of rule that are "interesting" in ways that elaborate comparative simplicity into extreme diversity, and I'd point to the quirky repetitiveness of many of our very standard rules of physics as evidence of just that.
A: If there are infinitely many possibilities shaping up our universe, then mathematically the probability that a specific outcome happens (say an a monkey reproducing Shakespear's work) is exactly ZERO! In physics, there are only probabilities and no certainties. Implications: the idea that an exact replicas of "me", and such, rests on a zero-probability event !
