Paradoxes and assumptions aside, is it literally true that there is a star at the end of every vector you can draw from my eyeball into space? I understand it's true if you assume the universe is infinitely large, but we know that's not true, right?
My understanding is that it would be true but for the fact that the longer those vectors get, the farther back in time they reach, and thus eventually reach into a time when stars did not exist and thus you hit the CMB instead. The stars you would have seen are in space that has moved away from us faster than c.
My understanding is that the universe is believed to be literally infinite in the three spatial dimensions that are familiar to us, and that its mass is also believed to be infinite. It really blows the mind.
Disclaimer: I'm a programmer not a physicist. :-)
EDIT: I did a little more reading on this. Answers are all over the place. But from what I can make out of the most recent sources, it seems that modern models treat the universe "as if" it were infinite although there is no way to know whether it is or not.
There are very few patches of the sky where if you look you won't see (very many) stellar objects or galaxies. Astronomers know about them--they're great for calibration of instruments.
That said, Hubble took a many-day exposure of one of these ultra-deep regions, and this is what it saw:
