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?
Answer
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.
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