Aquinas thought there were only a finite number of events in the past and that, since there are only a finite number of events in the past, there must have been a first event. Of course, from the existence of a first event, he wants to claim that there must have been an uncaused cause -- something non-physical which causally determines the existence of the universe (non-physical because he commits himself to the view that all physical things require causes. Thus, by Aquinas' lights, anything that is non-caused must be non-physical.) He identifies this non-physical thing as God.
This has me asking the following.
Do all events have a finite amount of time between them? If they do, then the universe can have some finite age. If they do not, it's possible to have a set of past events between which there is, at best, an infinitesimal amount of time. If this latter situation occurs, then adding together all past time intervals over a finite amount of events gives us a set of measure zero. In other words, no time would have past at all. There is a pretty good reason to think that the latter is the case. Under a certain view about what causation is, we can imagine Aquinas to be talking about each space-time slice causing the next space-time slice. That is to say, the conditions at t_n are responsible for causing those at t_n+1. Furthermore, that events are just the temporal slices. Objections pertaining to relativity are just complications; we can imagine constructing something like this from the vantage point of any inertial frame that you wish. Thus, if Aquinas commits himself to the view that there are a finite number of events in the past and to a certain view about what an event is, then we can construct a pretty strong reductio against him.
But Aquinas is really trying to argue that there must have been a first cause, not that there must have been a finite number of events in the past. If we take some closed subset of the real line, there will be a first element on the line. Thus, even with an uncountably infinite number of past events, we do not have to commit ourselves to the non-existence of a first cause.
However, this is problematic for the following reason. We could have just as well taken an open interval on the real line and mapped it to events. Explicitly:
Construct some finite open sub-set of the real line. Call this L. Now, map the points along L to points along the time line T representing the continuum of all past events. The successor relation on L corresponds to the causal relation along T. Since each member of L has a successor and there is no first member (by construction), all events on T have causes and there is no first cause.
The Hume-Edwards principle states:
If the existence of every member of a set is explained, the existence of that set is thereby explained. (From Pruss' "The Hume-Edwards Principle and the Cosmological Argument")Now, if Hume-Edwards is correct -- and I think it is -- then an adequate explanation for the universe can be given by citing the causes for each member of T in terms of some other member of T. Since there is no first cause on T, but each member of T has both a successor and a preceding element on T, an adequate explanation of the universe can be given without citing supernatural causes (provided that the universe envisioned in this thought experiment corresponds to our actual universe.)
Of course, we have no reason to think that there is no actual first event, but what this does adequately show is that a large space of possible defences of Aquinas are insufficient to establish the existence of a first cause.