If the universe were ageless, eternal, and unbounded in all directions (Einstein’s static universe model), every point in the night sky would terminate in a star, and thus the universe would be bright rather than dark; hot rather than cold. Edgar Allan Poe said it best in his 1848 essay Eureka — A Prose Poem:

No astronomical fallacy is more untenable, and none has been more pertinaciously adhered to, than that of the absolute illimitation of the Universe of Stars. The reasons for limitation, as I have already assigned them, a priori, seem to me unanswerable; but, not to speak of these, observation assures us that there is, in numerous directions around us, certainly, if not in all, a positive limit — or, at the very least, affords us no basis whatever for thinking otherwise. Were the succession of stars endless, then the background of the sky would present us an uniform luminosity, like that displayed by the Galaxy — since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all. That this may be so, who shall venture to deny? I maintain, simply, that we have not even the shadow of a reason for believing that it is so.

This is called Olbers’ paradox, and it is a thought experiment to explain the observed darkness of the night sky. If the conditions set above were true, we would see light in all directions; but we don’t, so it must not be. Olber’s paradox is often used to support theories of a finite universe, in which a finite number of stars radiate a finite amount of light in all directions — that is, the Big Bang theory, in which the universe has a set date on which it came into existence, and all effects can be traced backwards to it. However, there are a number of reasonable resolutions to Olber’s paradox — that is, conditions under which the universe might be infinite, unbounded, and eternal, without producing a brilliant night sky.

The number of stars in the universe may not be infinite. This is a common response — that, just because the universe is infinite, the number of stars (and, accordingly, the amount of matter) need not be infinite. However, this would imply that the density of the universe is zero — $1/\infty$, which is clearly an unphysical result, given the very real non-zero density of the objects around us.

Another objection is that there may be dust particles obscuring the view of the stars. This is another common response — it would make sense that the universe could be infinite but obscured, and that massive dust clouds around each star, or in the path of its light, could absorb and therefore dim the night sky. However, these dust clouds would also absorb heat from their stars, and thus would radiate energy proportional to their absorbed light. Even if they radiate a tiny fraction of their absorbed energy, a tiny fraction of an infinity of stars is still an infinity of light.

The most interesting objection is the notion that the structure of the universe may not be isotropic — that is, uniform in all directions. This would be a consequence of fractal cosmology, the theory that the structure of the universe varies wildly across scales; as in, the density and complexity of the universe, high at small scales, would decrease dramatically at larger scales. However: this argument, mostly a purely mathematical one, also violates the cosmological principle, a commonly accepted theorem stating that the universe is, generally, the same everywhere, in scale, structure, physical law, etc; that it is, in the words of William Keel, “playing fair with scientists.” This doesn’t rule out the model of a fractal universe, but it does reduce the support for it.

In the end, we generally accept the size of the universe to be more or less a moot point — modern developments in physics such as red shift have put limits on the size of the observable universe. Thus, it may be infinite and infinitely populated with stars, but the physical nature of the universe would make it impossible to observe any of them past a certain radius. This radius, a hard limit on the matter in the universe that we can see, would be equivalent to the speed of light times the age of the universe — in other words, any light from matter further away would not have reached us yet.