Can we use Euclid's second postulate ("A terminated line can be produced indefinitely") to say that the universe is infinite? Euclid's second postulate says > A terminated line can be produced indefinitely. Can we use this and say, Universe is infinite? I have read this post but if we consider universe as a three dimensional space, the universe will fall in to the postulate's domain (geometry).

Euclid's postulates were an attempt to represent the world as he experienced it, but of course experience does not include extending a line to infinity. I would guess (but I didn't ask him) he noticed that you could extend lines very far, assumed space was homogeneous, and made the postulate. From the wording it is not clear whether extending a line many times around a circle counts or not. If it does not, this one will rule out geometry on the surface of a sphere, where lines are great circles. I suspect his experience was not sufficient to distinguish the geometry of the surface of the earth from an infinite plane, even though there were famous Greek proofs that the world was round.