Is it consistent with beginning/ending principles for an interval-based temporal logic to cover a time that is exactly ω+ω intervals long?

Question

The SEP article on temporal logic reports on possible beginning/ending principles for instant-based temporal logics:

• beginning: ∃x¬∃y(y≺x)
• end: ∃x¬∃y(x≺y)

Note that ≺ is the prior-to relation, here. Now, from the look of things, it seems logically possible to assign any ordinal number, finite or transfinite, to overall lengths of such time, or to assign any cardinal number to overall quantities of instants or intervals. For obscure technical reasons that would be an unpleasant detour in this post, I am minded to work with an interval-based temporal logic to address a problem in a (partially toy) system that I'm judging. Is it logically possible to have an interval-based temporal logic with an ending factoid at its base, according to which the future length of time is exactly ω+ω (or ω+ω+ω or ω+ω+ω+ω or whatever) intervals?

Answer 1

No, any limit ordinal lacks an endpoint. In fact, the ordinals in which the two axioms hold are exactly the successor ordinals (e.g. ω+ω+1 is fine). Or maybe something like ω+ω*, where the * denotes the reversed order.