Finite Universe

finite number of stars

Olber's Paradox was originally thought of by Kepler.

from Encyclopedia Britannica:

"In 1610 Kepler provided a profound reason for believing that the number of stars in the universe had to be finite. If there were an infinity of stars, he argued, then the sky would be completely filled with them and night would not be dark! This point was rediscussed by the astronomers Edmond Halley and Jean-Philippe-Loys de Chéseaux of Switzerland in the 18th century, but it was not popularized as a paradox until Heinrich Wilhelm Olbers of Germany took up the problem in the 19th century. The difficulty became potentially very real with Hubble's measurement of the enormous extent of the universe of galaxies with its large-scale homogeneity and isotropy. His discovery of the systematic recession of the galaxies provided an escape, however. At first people thought that the redshift effect alone would suffice to explain why the sky is dark at night--namely, that the light from the stars in distant galaxies would be redshifted to long wavelengths beyond the visible regime. The modern consensus is, however, that a finite age for the universe is a far more important effect. Even if the universe is spatially infinite, photons from very distant galaxies simply do not have the time to travel to the Earth because of the finite speed of light. There is a spherical surface, the cosmic event horizon (roughly 10**10 light-years in radial distance from the Earth at the current epoch), beyond which nothing can be seen even in principle; and the number (roughly 10**10) of galaxies within this cosmic horizon, the observable universe, are too few to make the night sky bright.

Relativity makes it more complicated, but still results in a finite "observable universe".

"When one looks to great distances, one is seeing things as they were a long time ago, again because light takes a finite time to travel to Earth. Over such great spans, do the classical notions of Euclid concerning the properties of space necessarily continue to hold? The answer given by Einstein was: No, the gravitation of the mass contained in cosmologically large regions may warp one's usual perceptions of space and time; in particular, the Euclidean postulate that parallel lines never cross need not be a correct description of the geometry of the actual universe. And in 1917 Einstein presented a mathematical model of the universe in which the total volume of space was finite yet had no boundary or edge. The model was based on his theory of general relativity that utilized a more generalized approach to geometry devised in the 19th century by the German mathematician Bernhard Riemann.

...

"The principle of equivalence in general relativity allows the locally flat space-time structure of special relativity to be warped by gravitation, so that (in the cosmological case) the propagation of the photon over thousands of millions of light-years can no longer be plotted on a globally flat sheet of paper. To be sure, the curvature of the paper may not be apparent when only a small piece is examined, thereby giving the local impression that space-time is flat (i.e., satisfies special relativity). It is only when the graph paper is examined globally that one realizes it is curved (i.e., satisfies general relativity).

"In Einstein's 1917 model of the universe, the curvature occurs only in space, with the graph paper being rolled up into a cylinder on its side, a loop around the cylinder at constant time having a circumference of 2(PI)R--the total spatial extent of the universe.

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