Olbers started with an infinitely large space equably filled with stars like the Sun, our Sun is just one of the stars. He regarded the space around the Sun as divided into shells (e.g. of one light year thickness) one on top of the other. A star is said to be in a spherical shell if the star's center of gravity is in it. The number of stars in a shell varies with the square of the shell radius R. (This is not true for small R, but is ever better for larger R.) If radius R is twice as large, then the number of stars in the shell is four times as large. The brightness of the stars in a shell falls with R, is reciprocal to R squared. If the shell is twice as far away then the brightness of the shell is four times as small. The increase in brightness due to more stars is precisely canceled by the decrease in brightness because those stars are further away. As a result every shell lets fall the same amount of light on the surface of the Sun as any other shell of same thickness and center. Since there are infinite shells the sky at night should be infinite bright. This applies simultaneously to any other place in the universe like the Earth.
The universe that Olbers imagined in his time, starts from a number of assumptions, as we would state them nowadays. Assumption (1) is the Cosmological Principle, which says that, if you consider it at sufficiently large scale, the spatial distribution of matter in the universe at one specific moment is homogeneous (average matter/energy density is the same finite value everywhere). The described space of Olbers certainly meets this criterion. We disregard the clumping of stars into galaxies and the empty voids between groups of galaxies.
We neglect the 2.7 K background radiation that does not originate in stars (2) and also we neglect the Dark Matter (3) and Dark Energy (4). We neglect the existence of black holes (5). We assume that the universe is static, so there is no expansion of the universe, we neglect the big bang. All star velocities are assumed to be absent, like twin stars around each other, the larger movement of stars around the galaxy center and the movements of the galaxies themselves (6). The universe is infinitely large (7).
A thought experiment
Visible starlight originates from stars because star material is converted into light particles (photons) according to E = mc, which then escape from the star and go into empty space. We now perform a thought experiment. Suppose we convert, according to E = mc, all matter in the universe into visible light, all star matter, all planets and all gas and dust, everything that has energy and is not yet visible light. Now we have the maximum visible light density you can achieve with 10^-26 kg/m^3 (the average mass density of our universe). What remains then is a radiation bath or photon gas that is uniformly distributed all over of space, if you look at it on a sufficiently large scale. Compare this photon gas to the air in a large closed hangar somewhere on Earth. The molecules of the still air are not stationary at all but moving criss-cross through the hangar. Theoretically it could be that at a certain moment the movement of all air molecules in the hangar is such that they happen to gather all in one half of the hangar, the other half leaving vacuum, but statistics show that the chance for this is too low to take into account. So is the photon gas. Every sufficiently large piece of space contains the same amount of radiation and a certain area can only contain more radiation at the expense of neighboring areas which will then contain less radiation. On a sufficiently large scale, the probability for this is too small to occur. Remember that (part of) the radiation bath IS the starlight. If you were standing in the radiation bath, being the last remaining piece of material in the entire universe, and you opened your eyes you would always and everywhere see the same light intensity in all directions, no matter how much time would pass. The night sky in this thought experiment (it is always night then, you could say) then has a constant brightness and is therefore not infinitely bright at all. Due to several causes (not all dust and gas ever contract to stars, stars cannot burn up completely, the universe is expanding) the night sky that we observe on Earth is still less bright than that from our thought experiment. Olbers' paradox has in fact been resolved now.
Note that this solution is independent of the size of the universe and whether this size is finite or infinite. The solution is independent of the age of the universe or whether its life span is finite or infinite. It makes no difference to the solution whether the universe is expanding or not, although expansion will reduce the brightness of the night sky. All that matters is the average mass energy density.
Where is the error in the reasoning as formulated by Olbers? Let's try to imagine how Olbers saw the universe. All the stars were like the Sun, bright stars were closer suns and fainter stars were suns further away. All the suns were created and ignited simultaneously at the "Moment of Creation", a moment that we will call T = 0. The stars were evenly distributed over space and in that space they all stood still with respect to each other and with respect to our Sun, they would never leave the place of their origin. The space was infinitely large with an infinite number of stars in it. Olbers may have thought that the stars would shine forever, at least until the "End of Time". Olbers may have thought that starlight did not need time to go from a star to us.
We add to this picture two pieces of modern knowledge: the Sun (and therefore all other stars in Olbers' universe) have an estimated lifespan of 10 billion years and the speed of light is 3 x 10^8 m/s. We conclude that in the universe of Olbers all stars will die out simultaneously 10 billion years after T=0.
Realize that at moment T = 0 (actually 8 minutes after T = 0) on Earth only our Sun is visible. The other stars are already there, but their light has not yet reached us. In the course of the first 10 billion years, star after star becomes visible, the nearest stars first, then the nearest stars just behind them, and finally a sphere with a radius of 10 billion light-years is visible around the Earth full of stars.
Then, it's 10 billion years after T = 0, all the stars are put out at once. We don't see this. The only thing WE see at that moment (8 minutes after that moment) is that the Sun goes out. It is night from now on. But from that moment we also see stars disappearing over the years, the nearest stars first, then the nearest stars just behind them, etc.
Now think of the space around the Sun as divided into spherical shells of 1 light-year thickness centered around the Sun. Again a star is said to be in a spherical shell if the star's center of gravity is in it. As the stars in the nearest spherical shell go out, at the same time the stars in the spherical shell just outside the farthest visible spherical shell, light up for the first time. Those are not new stars, in fact those stars had already died now, more than 10 billion years after T= 0; the light of their ignition at time T=0 has only reached us now because of the great distance.
And this goes on forever. Olbers correctly noted that each spherical shell per unit time casts the same amount of starlight onto Earth as any other spherical shell of the same thickness and center. As a result, the brightness of the nearest spherical shell with visible stars (which is disappearing) and the brightness of the most distant spherical shell with visible stars (which is appearing) will be equal. As a result, the total brightness of the observed starlight will remain constant over time.
And that result agrees neatly with the result from the thought experiment. However, in the course of time the starlight that falls on Earth will be supplied by more and more stars at larger and larger distances. Even though the stars themselves are no longer there, their light still haunts until the end of time.
The mathematically formulated paradox of Olbers is reformulated here as a process of ever more shells becoming visible, one on top of the other, in doing so creating a sphere of increasing radius full of visible stars. However, what Olbers did not take into account is the disappearing again of shells because the stars in it had died and the light they used to send has ceased to exist. The visible entirety of stars is not a sphere but a shell, with a constant thickness of 10 billion years in this case, and receding from us with lightspeed. This is the error in the reasoning as formulated by Olbers.
Well, that is, the shell moves but the stars in it remain standing still. The shell maintains thickness but increases in volume and number of stars in it. As is argued, as long as the thickness of the shell remains 10 billion years, its luminocity as observed at the Sun and on Earth remains constant over time, no matter how far away the shell has proceded.
Also in the universe as we know it, there will come a day when all the gas and all the dust that can contract into stars, indeed has done so and all the stars that have been ignited finally have burned up. Let's for convenience set this at 10^12 years after T = 0 (just a number, no try to be accurate!)
Different stars have different lifetimes. Also that lifetimes lie at different places along the time axis. When one includes these two properties and still wants to perform the paradox solution, one can in mind elongate each star lifetime such that it stretches from T = 0 to T = 10^12 years after T = 0. The entire amount of visible light a star radiates during its lifetime is thought to be spread out over these 10^12 years. The total output of light from the stars remains unchanged, only the stretch of time in which it occurs, changes. Then perform the solution. The visible entirety of stars is now a shell of 10^12 lightyears thickness, receding at lightspeed. After performing reduce the stretched-out star lifetimes back to their original position in time.
It takes a longer time but the end result is still the same universe of dead stars and a radiation bath of starlight that haunts it forever.
Reviving Olber's paradox
As happens more often, when one has become used to the presence of a riddle, one feels some grief in being obliged to take leave. After a few hundred years the paradox of Olbers is no more. However, for those people I have news: we can revive the paradox! Just set the speed of light at infinite and then at T = 0, the moment where in Olbers' universe all stars start to shine, at each point of the universe will immediately arrive the first photon from each star. And since there are infinite stars in Olbers' universe, there will be, from the Moment of Creation, an infinite number of photons at each possible place. The sky at night will be infinite bright. Again.
When c is infinite, the magnetic field disappears, and so will all special relativistic effects like time dilation and length contraction. The Sommerfield fine structure constant is set to zero. That universe bears little resemblance to ours. But in principle there seem to be no compelling arguments against it, physically it seems to be a possible world.
E(Sun) = m(Sun) * c (from special relativity, due to the c occuring in the formula),
E(Sun) = the energy of the Sun, m(Sun) = the mass of the Sun (and thereby the mass of every other star in Olbers universe) = 1.99 * 10^30 kg, c = speed of light = 3 * 10^8 m/s.
E(1 photon of yellow light) = h * f (from quantum mechanics, due to the h occuring in the formula),
E(1 photon of yellow light) = the energy of 1 photon of yellow light, h = constant of Planck = 6.63 * 10^-34 Js, f = frequency of the yellow photon = 0.51 * 10^15 Hz.
The energy of the Sun can be set equal to N times the energy of one photon of yellow light.
E(Sun) = m(Sun) * c = N * E(1 photon of yellow light) = N * h * f
1.99 * 10^30 * (3 * 10^8) kg*m/s = N * 6.63 * 10^-34 * 0.51 * 10^15 J*s/s
N = 5.30 * 10^66
So the Sun can be thought of to consist of (less than) 10^67 photons of yellow light.
Our real Sun consists of a lot of other particles than photons alone: protons and neutrons, eventually combined to nuclei; some antiprotons and antineutrons; electrons and some positrons. Without help from the outside these particles never can entirely convert to radiation. So for real suns (one solar mass, not the very heavy stars) the number N is rather 10^63 than 10^67.
When c is infinite then one moment the photon is still part of its origin, the source particle, and the next moment the photon is the straight line connecting source and goal. And the moment thereafter the photon is gone already, absorbed as it is by the goal particle. When c is infinite, the photon exists only one single moment as a straight line segment in curved space. There is not one moment where photon is a particle as we know it in our universe.
For an observer that wants to see a specific star there are lesser than 10^63 photons available. Only a limited number of observers will ever see one single photon from the star. There are not enough photons in our star to make the star visible for an infinite number of observers. Compare the photon gas from
A thought experiment above. When c is infinite, the emission of photons by stars is merely a homogeneous redistribution of star matter/energy over the universe. The mass density of the universe was finite and every star redistributes likewise by starlight emission, so the average matter density will change very little over time in the cube lightyears areas of the universe. The sky will not be infinite bright. When c is infinite, the brightness of the sky will reach a certain upper limit and then will remain constant forever. This is of course except for the expansion of the universe.
A special case
Imagine an Olbers universe at a certain moment T = t, with light speed restored at well known 3 * 10^8 m/s, but with an infinite long past preceding moment t. Suppose over the spacetime diagram of that universe there is a homogeneous distribution of stars being born, live and after 10 billion years die again. There is no moment T = 0 anymore where all the stars are created at once. Since star creation goes on all the time everywhere, the remnants of the dead stars remain (dead star bodies and the radiation of an entire star lifetime) and the density of remnants at T = t will be infinite. It is no surprise the sky will be infinite bright then.
However, regard in this universe only the past light cone of the Sun at moment t in the spacetime diagram. If one takes all space around the mathematical past light cone for, let's say 10 lightyears around each point of the past light cone, then such a cone layer will be filled with stars which light all happen to arive at the Sun at the same moment t. And since the cone stretches infinitely far into the past, the amount of light simultaneously ariving at the Sun's surface will be infinite.
Suppose we would be able then to erase all stars except for the stars in that past light cone layer. The result is a special kind of finite-density universe, resembling the original Olbers universe, where during about one star life time the sky around the Sun (and around the Earth too) will be infinite bright. Olbers paradox again, albeit for one very special kind of universe.
It is about the amount of light arriving at the Sun and the Earth at a certain moment t in time. It is not important how precisely that light happened to cross that space. Precisely the same amount of light will arive at the Sun, and the Earth, in the past-light-cone-universe at moment t as it does at moment t in the infinite-lightspeed-universe described in
Reviving Olbers paradox above.
Space can be divided in neatly connecting shells around the Sun of e.g. one lightyear thickness (or sufficiently smaller). In the past-light-cone-universe shells further away are further in the past, while in the infinite-lightspeed-universe shells are all simultaneous in the present. But in both cases it are the same shells, same dimensions, distance and thickness that makes their light to arrive at Earth and the Sun at moment t. In the past-lightcone-universe the light comes from the past, while in the infinite-c universe the light comes from the present, but the arriving amounts at a certain moment t are the same.
Since the paradox is solved for the infinite-lightspeed universe, it must be solved for the past-lightcone-universe too. Of course provided the expansion of the universe is set to zero.
This encourages the opinion that from all possible paradoxes of Olbers, if I only knew them, I would solve them all. Well, I hope the details of all those special cases will not be too difficult.