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In an infinite universe...would there be infinite 'me'?

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This question might seem ridiculous at first, but it follows a process of thought. The other day I was talking to a professor and he believed that if the universe is infinite, not only will there be another 'him', there will be infinite copies of him. The reasoning: "the probability of another 'me' existing in the universe is not zero, even if it is seemingly very low. So as the number of trials tend infinity, the probability of at least a single copy of 'me' existing again increases and increases. Eventually, we reach a number of trials/'spaces' that the probability of at least one other of me existing is so high...there will have to be another copy of me certainly. Put another way, matter can arrange itself in different ways (quantum states). But the number of different arrangements is not infinite. So after a certain amount of space, the combinations will start to repeat over again. I (as a person) and my entire life (as a sequence of events) ...