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First of all, let me tell you that I'm not a physicist but rather a computer scientist with a mere interest in physics at nowhere near a professional level so feel free to close this question if it doesn't make any sense.

I remember a physicist friend mentioning me about an argument about the finiteness of the universe. I have looked it up and it turned out to be Olbers' Paradox.

We computer scientist like to use astronomical numbers to help us imagine the complexity of an algorithm. One of the most common one is the number of atoms in the Observable Universe (which we take as $10^{80}$) so I have a crude understanding about the observable universe concept.

I had known these two for some time hence I woke up with the dilemma today. So my question is, how come it can be argued that universe is finite just because it is dark if we know that we can only observe a finite portion of it? Can't it be the case that the universe is infinite even if the sky is dark because not all the light from all the stars reach the earth?

I have searched this a little bit but I think I need an explanation in simpler terms (like popular physics). A historical perspective would also be welcomed.

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People answering here, please remember, when stating the paradox, that light from distant sources goes down as the square of distance. –  arivero Dec 9 '12 at 22:19

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So my question is, how come it can be argued that universe is finite just because it is dark if we know that we can only observe a finite portion of it?

You are mixing two theories here. Olbers paradox has as a basic theory a static infinite in space and time universe. The dark night sky means that either the universe is not static, or has a beginning, or has a finite extent in space. Or all three.

Can't it be the case that the universe is infinite even if the sky is dark because not all the light from all the stars reach the earth?

A different model than a static infinite in space and time universe is needed in this case, an infinite universe that appeared at a time t=0, for example, so that the light of distant stars would not have reached us by now. But there are more data than the dark night sky to be fitted by a cosmological model and the available data fit the Big Bang model quite well:

the Big Bang occurred approximately 13.75 billion years ago, which is thus considered the age of the Universe. After its initial expansion from a singularity, the Universe cooled sufficiently to allow energy to be converted into various subatomic particles, including protons, neutrons, and electrons.

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Olber's paradox just states that if the universe is infinite and static (and flat) then any line of sight to the sky will invariably hit a star/source of light, and therefore our sky should be bright all the time.

The observable universe is a concept that arises from the expansion of the universe. Because the universe is expanding in such a way that points farther away from us are moving away faster than points closer to us (i.e, the further something is from us, the faster it's moving away) there is a definite point beyond which light will never reach us, since that point is moving away faster than the speed of light.

Now to come to your question - You're absolutely right. We don't actually know whether then whole (observable + the rest) universe is finite or not. There are theories which hypothesize the lower limit on the size of the entire universe (not just the observable universe) but in literature people mostly restrict themselves to speaking about the observable universe since we cannot (by definition) observe anything outside the observable universe, so no hypothesis can be tested.

This isn't necessarily a contradiction of Olber's paradox, since he was talking about an infinite and static universe. So his paradox can be seen as favouring an expanding universe model rather than a static one to explain what we see.

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Here is an analogy for you:

You find yourself in a forest. A forest ranger stand next to you. He tells you the forest goes on and on and has no boundary. You look around. There is tree trunks all around you, yet you do see gaps between the trunks. How do you react to the forest ranger's remark? You tell him he is wrong. If the forest was infinite in extent, wherever you look, your line of sight would ultimately hit a tree trunk. The forest ranger thinks for a while and responds: "the forest stretches without limits, but you only see part of the forest. This is because the ground is not flat, and distant trees disappear behind the horizon."

As any analogy, this is not a full representation of 'the real thing'. It does not provide an analogy for all aspects of the commonly accepted solution to Olbers' paradox. Still it might help explaining how concepts like "observable universe" contribute to the resolution of the paradox.

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+1 for the beautiful analogy –  gokcehan Dec 11 '12 at 11:02

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