The traditional Third Way was meant to show that a necessary entity exists by positing a universe eternal in the past.
1. Every existing entity is either contingent or necessary. (Definition)
2. Something exists now. (Premise)
3. If something exists now, then something or other has always existed. (Premise)
4. Hence, something or other has always existed. (From 2 and 3)
5. The past is infinite. (Assumption)
6. Given infinite time, all potentialities will have been actualized. (Premise)
7. The concurrent non-existence of all contingent entities is a potentiality. (Premise)
8. Hence, the concurrent non-existence of all contingent entities has been actualized. (From 5, 6 and 7)
9. Therefore, a necessary entity exists. (Implied by 1, 4 and 8)
Now, what if we assume the negation of (5), keeping the (1) through (4) the same?
5'. The past is finite. (Assumption)
6'. Given finite time, the concurrent non-existence of all contingent entities has been actualized. (Premise)
7'. Therefore, a necessary entity exists. (Implied by 1, 4 and 6').
The new debate would probably center around (6'). Could a contingent entity exist at t0, a sort of undifferentiated time? A modalized version of the Third Way would be immune to such an objection, since it's at least possible that nothing contingent existed at t0.