The inner workings of the Olbers paradox A long time ago I was told that the universe is finite. The provided "proof" (or reasoning), known as Olbers' paradox, was that on infinite universe there would be an infinite number of stars, and that an infinite number of stars in night sky would make the sky bright.
I guess this sounds reasonable, but I have some doubts. If we assume that universe really is infinite and that it has infinite number of stars spread somewhat uniformly on the space, how would one prove (or disprove!) that the night sky really would be bright? What assumptions one would need to make on the way constructing the proof?
I am not looking for a "correct" answer but how to approach the claim mathematically.
 A: I have nothing to say about the (possible) infinity of the universe; however, it is the case that infinitely many stars distributed uniformly will create infinite brightness at any point.
Let's throw away all the real physical facts about stars, and make the (incorrect, but good enough for now) assumption that the universe is an infinite volume with some 'star' distributed uniformly throughout it.  So there is a uniform 'brightness field' which I'll denote by $\phi$.
Now suppose we are standing at the point $\mathbf 0$.  How much brightness are we getting from a small volume $dV$ at a distance $r$ from us?  Well, the small volume is emitting light rays at an intensity of $\phi dV$, but as you get further away (say, at a distance $s$) from the volume, the light rays are spread over the surface of a sphere of radius $s$.  The surface area of a sphere of radius $s$ is proportional to $s^2$, so the light intensity due to the volume at a point a distance $r$ away from the volume is proportional to $\frac{1}{r^2}$.

(source: ohio-state.edu)
So the volume $dV$ contributes a brightness of $\frac{C}{r^2}dV$ to the total brightness, where $C$ is some constant, and $r$ is the distance of the small volume from $\mathbf 0$.  We are now in a position to integrate over a large sphere of radius $R$ to get the brightness due to all the 'stars' at a distance less than $R$ from us:
\begin{align}
\textrm{Total brightness at distance less than $R$} &= C\int_{|\mathbf x|<R}\frac1{|\mathbf x|^2}dV\\
&=C\int_0^{2\pi}d\phi\int_0^\pi \sin(\theta)d\theta\int_0^R\frac1{r^2}\times r^2dr\\
&=C\times4\pi\int_0^R dr=4\pi CR
\end{align}
So the brightness due to the stars in a sphere of radius $R$ is proportional to $R$.  Clearly, if we make $R$ infinitely large, the brightness will become infinitely large as well.
Of course, this ignores several physical realities, such as the expansion of the universe, the geometry of space-time, etc.  But that is, at least, the mathematical justification for your friend's claim.
A: There is one thing that you have failed to consider: the expansion of the universe. Since the expansion of the universe is "faster" the further away it is from us, there will come a point where the expansion is faster than the speed of light. (This actually happens, I'll add sources if I have the time to do so.) So, the light from stars at or greater than that distance would not be able to reach us.
Secondly, since infinite time has not elapsed, not all light have reached us.
Edit: great video to answer your question
http://www.youtube.com/watch?v=gxJ4M7tyLRE
A: There are things that block out brightness (relative brightness). The sun is the brightest star during the day, and most stars are way too faint and distant to notice them as bright.
A: If all photons decay after some arbitrarily large time period, Olbers paradox is no longer an objection to the proposed infinite extent of the universe.
