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So I've been thinking about the infinite universes model, where each possible action or event creates a new universe for each outcome. For example, if you flip a coin there will be one universe in which the coin came up heads, and another in which it came up tails. Each new universe then has an entirely new set of possibilities which all spawn yet more universes.

So I was thinking about a machine in which you could simulate the entire universe from beginning to end, and seeing the consequences of each result. Of course there are infinitely many, but if you had perfect information you could pick out your universe by looking at the past. Basically if you looked at it as a tree, where each branch was a new possibility, you could find your branch by following from where it grew.

My question comes when you start trying to look at the future. There are an infinity of possible futures, and this machine could certainly predict one of them, or many of them, but it's very unlikely you'd ever predict the one you are on.

So this has been more philosophical than mathematical, but now I come to my question:

Is it mathematically possible to guess the correct branch in an infinity of choices? Or put another way, if I had an apple, and I threw it into a barrel filled with an infinite number of apples, could I ever correctly guess which one was mine? Is it possible to point to one and correctly say, "I think that one is mine"?

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It's a tricky concept. The problem is that you're trying to force the physical concept of "guessing (and verifying)" into a purely mathematical realm. If I choose a number at random from the real numbers, the probability that it is a rational number is zero. Does that mean I could never find a rational? We know they exist, so certainly not... The probability is zero but the event space is non empty. It happens. –  Tyler Mar 3 at 19:15
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See this thread –  Tyler Mar 3 at 19:16
    
To expand slightly on Tyler's point, suppose we pick a real number "uniformly at random" between 0 and 1. There are infinitely many possibilities. It is certain that some number will be chosen, but the probability of any individual number being chosen is zero. So the choosing of (say) one-half exactly is just as possible as choosing any other number in that interval, while the probability of choosing one-half is zero. –  hardmath Mar 3 at 19:57
    
Note that there are infinitely many possibilities for the past too, even after you take into account what you actually remember happening. There is some merit in doing away with the notion that "the universe you are actually on" makes sense, and instead view the universe as something that lets you answer questions like "given facts X and Y, can Z be true?" (whether Z is about the past, present, or future). This is more interesting when you allow probabilities rather than just true/false. This is even more interesting in the setting of quantum physics. –  Hurkyl Mar 3 at 20:04

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This may be very counterintuitive, but if you accept the axiom of choice, it is possible to design a method that predicts the future with probability 1. See this amazing paper. However, it's non-constructive, so although mathematically one may do this, you can't program a computer to do so.

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