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 or large shell thickness, but is ever better for larger R or thinner shells.) If radius R is twice as large, then the volume of the shell and thus the number of stars in it, 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.
Every shell around the Sun lets fall the same amount of starlight on the surface of the Sun as any other shell of same thickness and center. (Olbers Argument)
Don't take the shells too thin or with too small radius. Since there are infinite shells one on top of the other, the sky at night should be infinite bright. This applies simultaneously to any other place in the universe like e.g. 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).
You can see the sky at night is not infinite bright. Instead it is relatively spoken quite dark. 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. They all shine with the same constant luminocity from their birth to their death. 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:
1) The Sun (and therefore all other stars in Olbers' universe) have an estimated lifespan of 10 billion years, and
2) 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 on Earth only our Sun is visible. (Actually 8 minutes after T = 0, that is the time light needs to travel from the Sun to the Earth.) 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 Sun and 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.
The 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. In the course of time the starlight that falls on the Sun and the Earth will be supplied by more and more stars at larger and larger distances. 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. Even though the stars themselves are no longer there, their light still haunts until the end of time.
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 Olbers paradox solution, one can in mind elongate each star lifetime into the past as well as into the future, 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. 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 constant density of starlight that haunts it forever.
Another Solution to Olbers' Paradox
We start at our mentioned
10^12 years after T = 0 when all dust and gas in the universe that can contract to stars indeed has done so and all stars that had been formed this way had radiated all the light of their entire lifetime.
All the light of their entire lifetime is per star a large but finite number of photons, a large but limited amount of radiation (see next paragraph). What remains then, apart from dead star bodies, 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. Mind the starlight IS the photon gas. From the gas little is lost anymore (dead star bodies will absorb a tiny little and most of it is re-emitted again) and nothing will add to it anymore. Every sufficiently large piece of space of a certain volume contains the same amount of radiation and a certain area can only gain more radiation at the expense of neighboring areas which then will contain less radiation. On a sufficiently large scale, the probability for this is too small to occur. In an Olbers universe the radiation density will be kept at constant and that's all there is to the light of the stars.
If you were standing in the radiation bath you would always and everywhere see the same light intensity in all directions, no matter how much time would pass. The night sky (it is always night then, you could say) then has a constant brightness and is therefore not infinitely bright at all. Olbers' paradox is resolved, without using the Olbers Argument, and the result fits in neatly with the result from the previous paragraph.
In first instant (one can always develop complicated universes where the arguments put forward here are not completely valid) this solution is independent of the size of the universe or 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 that the average mass energy density exists and is a finite value, which nowadays in our universe (not Olbers') is set at 10^-26 kg/m^3.
The solution in brief: imagine all mass-energy in the universe via E = m * c being converted to radiation, for convenience radiation of yellow light photons only. This is a photon gas with from then on constant density. It cannot become worse, there is no energy available anymore except the photon gas. The brightness of the night sky will be the brightness of the photon gas, a finite and constant value in an Olbers universe.
Realize that the 2.7 K remains of the Big Bang is an extra value added to the photon gas from the stars. The resulting value is a higher light density than with the starlight photon gas alone but the reasoning made so far doesn't change. Both the 2.7 K and the starlight then diminish in brightness due to the expansion of the universe. OUR universe of course, the Olbers universe has no Big Bang and thus no remains of it.
Most of the very distant stars are invisible
Very distant here means much larger than the about 1,4 * 10^10 lightyears which is the radius of our visible universe, nowadays. Visible starlight originates from stars because star material is converted into light particles (photons) which then escape from the star and go into empty space. For the emitted photons and so the radiation bath of the previous paragraph to exist, it is useful to realize that this means the addition of another two pieces of new knowledge to the universe of Olbers.
3) E = m * c (from special relativity, due to the c occuring in the formula).
In this case we take E(Sun) = m(Sun) * c.
Herein is 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.
4) E(photon) = h * f (photon) (from quantum mechanics, due to the h occuring in the formula).
In this case we take E(1 photon of yellow light) = h * f (1 photon of yellow light).
Herein is E(1 photon of yellow light) = the energy of 1 photon of yellow light, h = constant of Planck = 6.63 * 10^-34 Js, f (1 photon of yellow light) = 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
If we only regard the mass-energy, 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. It turns out that for real suns (one solar mass, not the very heavy stars) the number N is rather 10^63 than 10^67.
We are now in an Olbers universe enriched with 4 new pieces of knowledge. For an observer on Earth, orbiting our radiating Sun, that wants to see a specific star, there are (less than) 10^63 photons available from that star. Only a limited number of observers will ever see a single photon from the star. There are not enough photons in our star to make the star visible for an infinite number of observers.
In the visible part of OUR universe 10^63 is quite a number of photons. It is enough to make visible even the most distant stars, the ones that had formed just after the Big Bang. But in an infinite static Olbers universe - enriched with 4 new pieces of modern knowledge - virtually all very very distant stars never will have emitted even one single photon that crosses our solar system.
A very peculiar universe
In the universe of Olbers - to which two new pieces of knowledge were added - all the stars are born at the same moment (our Sun is one of them) and they all die at the same moment. Let's reshape Olbers' universe in a very peculiar way to regain our lost state of paradox. We now state that a star at e.g. 4.3 ly distance from the Sun is born 4.3 years earlier than the Sun (and also dies 4.3 years earlier than the Sun will do). A star at 11.5 ly is born 11.5 years earlier and a star at 100,000 ly is born 100,000 years earlier, and so on. There is no moment T = 0 anymore, instead there is an infinite long past preceding moment T = t, where our Sun lives. This universe is constructed in such a way that no matter how far away a star is, its light will always reach us at moment T = t. As a consequence, during its entire lifetime, from the birth of the Sun until its death, the light of all stars in the entire peculiar universe reaches us at the same time now, instead of nearer suns being visible earlier and suns further away being visible only later. Since space as well as time-into-the-past is infinite in the peculiar universe, the amount of suns which light reaches us simultaneously, is infinite too. The sky must be infinite bright, again.
The original Olbers universe can be divided in thin shells of same thickness around the Sun, all shells neatly one on top of the other, no overlap, no gaps. Not TOO thin of course, there must be a sufficient number of stars in it. Each shell shines the same amount of star radiation per unit of time on the Sun and the Earth (Olbers Argument). For each shell in the original Olbers universe there is a corresponding shell in the peculiar Olbers universe, enveloping precisely the same stars. That you have to take a shell at 100,000 ly distance in the peculiar universe 100,000 years in the past is not important. The Olbers Argument has no dependence on time. *) And since in the original Olbers universe all shells shine the same amount of starlight onto the Sun and the Earth, all the corresponding shells in our peculiar universe will do the same. And in the peculiar universe they will do that all simultaneaously. And there are infinite of such shells, so I cannot come to an other conclusion that as long as the Sun lives in this special universe, the sky indeed will be infinite bright. So yes, the reasoning is Olbers-like since it is counteracted by the Olbers Argument. But no, it is not a paradox.
*) The corresponding shell in the peculiar universe is not precisely the same as the one in the original universe: in the peculiar universe the suns at the outer side of the shell are a little older - born earlier - than the suns at the inner side of the shell. However, it are the same stars at the same positions and all of them shine with constant brightness all their life. So if you stay away from the edges, the moments of birth and death of the stars in the shell, you won't notice the difference.
Infinite light speed: reviving Olbers' paradox
As happens more often, when one has become used to the presence of a riddle and then the riddle is solved one feels some grief in being obliged to take leave. After a few hundred years the paradox of Olbers is no more. However, 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 Earth 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 reaching Earth at the same moment. 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, length contraction and E = m * c. The Sommerfield fine structure constant is set at zero. That universe bears little resemblance to ours.
I cannot use the a-star-equals-10^63-photons picture as a solution because SR has been thrown out and so E(Sun) = m(Sun) * c doesn't hold up anymore. Maybe I still can bring in the photon gas argument to solve it. For every area like e.g. the solar system, holds that what goes in has to go out. The emission of photons by stars, even when those photons have infinite speed, is merely a homogeneous redistribution of star matter/energy over the universe. IF the photon particle picture holds up when lightspeed is infinite.
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 the photon is a particle as we know it in our universe.
I guess E(photon) = h *f still holds. But what is the frequency of an infinite speed photon? What is the wavefunction of an infinite speed photon? Does the photon still have an impulse? Then I guess, when you have photons connecting two spots A and B, some of them transfer impulse from A to B while the rest must transfer impuls from B to A.
Maybe you arrive at a coherent world when you let approach c to infinity while you simultaneous let approach h to zero, e.g. by keeping the product hc at the constant value the product has now. See a tiny little more about it in The Light Speed - Planck Constant - conjecture, the last 10 lines about the Sommerfield fine structure constant at page 6 of the storyline QCD.
I am affraid I have to give in. I don't have the skills to treat infinite speed photons.