[...] parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series must have a final member. Anyone who thinks that an end-less series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite. [Wes Morriston: Beginningless Past, Endless Future, and the Actual Infinite] http://www.academia.edu/647482/Beginningless_Past_Endless_Future_and_the_Actual_Infinite
Endless for ever or endless at the moment are two different things. The time passed since the big bang is not endless. The time to pass may be endless. But the passed time will never report "infinity reached".