[Aengeuad]

The paradox itself is that if the universe is both infinite and eternal then we shouldn't have a dark sky. This is 'trivially' solved by demonstrating that either of those conditions aren't true. The article opts to use the explanation given by Edgar Allan Poe to demonstrate this, which is that the universe has a finite age and the speed of light is finite so there's only a finite amount of observable universe, which gives us a universe which was darker in the past and one that will only get brighter in the future as more of the universe becomes observable. This has some problems of course and the model Poe would have been working with would have been one of a cyclic universe of eternal growth and decay. This leads us to the Big Bang theory.

>> The redshift hypothesised in the Big Bang model [...]

>Doesn't sound to me like it's more than a hypothesis, but I could be wrong.

The way the Big Bang theory resolves the paradox is similar to that of how Poe resolved it, with a finite cap on the age of the universe there's only a finite amount of observable universe, and similar to Poe's explanation it presents a problem in that a younger universe would have been immensely bright. However this new issue is resolved through the explanation of the expansion of space which can be observed through the redshift of distant galaxies.

As we do observe a dark sky we know the hypothesis that led to the paradox can't be true, namely that the universe is both infinite and eternal, so the question is less about why we have a dark sky and more about what possible alternate hypotheses resolve the paradox. While the Big Bang theory is just a theory it's important to remember that proof is reserved for maths, a theory is a hypothesis backed up by observational data. General relativity led to the hypothesis of an expanding universe and this was something that was later observed from redshift measurements and from it we derive the Hubble–Lemaître law, that galaxies are moving away from earth with speeds proportional to their distance, in some cases faster than the speed of light, this alone fully resolves the paradox and crucially the Big Bang theory is not incompatible with this observation.

[vba616]

>The paradox itself is that if the universe is both infinite and eternal then we shouldn't have a dark sky

I've never really felt like I understood the paradox, since the way people explain it, it sounds to me like they're just denying the idea that an infinite sum can have a finite value.

Like, why couldn't the brightness of the sky in an infinite universe be any value at all, depending on the density?

The argument against an infinite universe that makes sense to me is that it would collapse on itself. But as a thought experiment, the stars could be massless and/or fixed in place.

[ravi-delia]

I mean, it's not like some abstract denial of convergent sums. You can work out the sum yourself, it doesn't converge. The density doesn't matter if it's consistent through the universe (which is one of the possible outs, but our universe is indeed consistent on the very large scale). The article explains it pretty well, and it's not hard to formalize:

> To show this, we divide the universe into a series of concentric shells, 1 light year thick. A certain number of stars will be in the shell 1,000,000,000 to 1,000,000,001 light years away. If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell, which is between 2,000,000,000 and 2,000,000,001 light years away. However, the second shell is twice as far away, so each star in it would appear one quarter as bright as the stars in the first shell. Thus the total light received from the second shell is the same as the total light received from the first shell.

The argument that the universe would collapse in on itself is made using similar math (gravity decreasing with the square of the distance is by no accident the same as brightness) so if you buy one you sorta have to buy the other. Of course, the universe probably is infinite and isn't collapsing in on itself, but that's because of dark energy (OK yes, there are all sorts of universes that obey relativity, and some of them are infinite and not collapsing, but if we live in one of those no one's found the solution that fits our observations. The dominant thinking was that the universe would eventually collapse until we discovered it's actually expanding).