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I was just wondering: if the universe is really infinite, and there is a certain probability to find a life form just like me on another planet (for example $1.0 \cdot 10^{-150}$), is it therefore certain that a life form exists which exactly looks like me at this moment?

You could think: $\text{very-small-number} \cdot \infty >1$

Just interested on your thoughts on this.

Any help is appreciated.

bashoogzaad
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  • No. The universe is by no means infinite - it has finite mass, and finite spatial extent. Big, but finite. I think the probability of another you in the universe is much closer to zero than to one. – Floris Oct 08 '14 at 12:16
  • The observable universe is finite. The universe? We don't know, but it appears to be flat, and that means either a rather weird geometry or infinite. – David Hammen Oct 08 '14 at 12:29
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    There's a hidden flaw in this question: It assumes a uniform probability over an infinite probability space. It doesn't matter if the space is the set of integers, the reals, or $\mathbb{R}^N$. You can't do that. It's invalid. – David Hammen Oct 08 '14 at 12:32
  • Thanks @Floris and DavidHammen for your comments. I see that the uniform probability assumption is not correct. – bashoogzaad Oct 08 '14 at 13:35

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As mentioned in a comment, the observable universe is finite. However, it is reasonable to think that the total universe may be infinite. Although we don't know this for certain, there is no experimental evidence that definitively contradicts that (eg. the curvature of space is very close to zero, within the precision of our measurements).

In fact, Brian Greene, in his book "The Hidden Reality" discusses the implications of an infinite universe. If the universe is truly infinite, then in the far reaches of space, there are regions where configurations of matter necessarily repeat itself, because there are only a finite number of ways that a given number of particles can combine. He then estimates the number of subatomic particles in the solar system, I believe $10^{80}$, and then calculates, an average distance in which we could expect to find a duplicate solar system. The distance is truly gigantic, and well outside the bounds of our observable universe. For this reason, the question almost becomes trivial, because even if the universe is infinite, and our solar system is duplicated somewhere, it is very far outside our light cone, and therefore we could never interact with it.

Sean
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  • Thank you very much for this answer. It is totally clear to me now. – bashoogzaad Oct 08 '14 at 13:32
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    Greene made the same fundamental error in that calculation that bashoogzaad made in his question. Greene implicitly assumed a uniform distribution across an infinite universe. That's not a probability. The measure is infinite rather than one. That violates a fundamental postulate of what "probability" is. A probability measure is some quantity whose measure over the entire sample space is identically one. – David Hammen Oct 08 '14 at 14:36
  • I agree with David Hammen. Even excellent physicists have an unfortunate tendency to make fools of themselves as soon as they publish for books written for laymen. Determinism in physics does NOT postulate that the same configuration will lead to the same result after finite time and since the number of possible physical histories far outnumbers the number of configurations (even in Greene's toy model of a universe), the probability of finding the same history is, essentially, zero, even based on his poorly motivated statistical argument. – CuriousOne Oct 08 '14 at 21:16