In “Mathematics of Eternity Prove The Universe Must Have Had A Beginning” (Physics Arxiv blog, April 24, 2012), KFC tells us, “Cosmologists use the mathematical properties of eternity to show that although universe may last forever, it must have had a beginning”:
Audrey Mithani and Alexander Vilenkin at Tufts University in Massachusetts say that these [current cosmology] models are mathematically incompatible with an eternal past. Indeed, their analysis suggests that these three models of the universe must have had a beginning too.
Their argument focuses on the mathematical properties of eternity–a universe with no beginning and no end. Such a universe must contain trajectories that stretch infinitely into the past.
However, Mithani and Vilenkin point to a proof dating from 2003 that these kind of past trajectories cannot be infinite if they are part of a universe that expands in a specific way.
They go on to show that cyclical universes and universes of eternal inflation both expand in this way. So they cannot be eternal in the past and must therefore have had a beginning. “Although inflation may be eternal in the future, it cannot be extended indefinitely to the past,” they say.
They treat the emergent model of the universe differently, showing that although it may seem stable from a classical point of view, it is unstable from a quantum mechanical point of view. “A simple emergent universe model…cannot escape quantum collapse,” they say.
More.