I like to update these arguments, especially when I think the phrasing can be improved so as to avoid confusion or disagreement. Here is what I currently have:
1. Every existing being is either temporally contingent or temporally necessary. (Definition)
2. Something exists right now. (Premise)
3. Necessarily, if something exists right now, then something has always existed. (Premise)
4. Possibly, there was a time in the past at which nothing temporally contingent existed. (Premise)
5. Therefore, a temporally necessary being exists. (Conclusion)
6. Every existing being is either omnipotent or non-omnipotent. (Definition)
7. Possibly, whatever is non-omnipotent can be generated. (Premise)
8. Necessarily, whatever is temporally necessary cannot be generated. (Premise)
9. Therefore, a temporally necessary and omnipotent being exists. (Conclusion)
Validity of the Argument
Assume (10): A temporally necessary being does not exist. (10) and (3) imply together with (1) and (2) that (11): Necessarily, a temporally contingent being has always existed. This contradicts (4), so (10) is false. Therefore, a temporally necessary being exists. Let us call this being "N."
Assume (12): N is non-omnipotent. (12) and (7) imply (13): Possibly, N can be generated. However, (13) contradicts (8). Therefore, N is omnipotent. Q.E.D.
Soundness of the Argument
(1) is true by definition. (2) is true upon observation. (3) is based on ex nihilo nihil fit (out of nothing comes nothing). If there were ever a time in the past at which nothing existed, then nothing would exist now, which is plainly false.
(4) seems reasonable enough. If one part of a house can fail to exist at some time, then the house as a whole can also fail to exist at some time. Given the possibility of the non-existence of some temporally contingent being at some time in the past, it seems equally possible for nothing temporally contingent to exist at some time in the past. Mind you, this does not assume there actually was such a time, but only that it is possible.
(5) logically follows from (1) - (4), so we need only close the gap between N and God in order for the modal third way to be a sound argument of natural theology.
Once again, (6) is true by definition. (7) might be the most controversial premise of the argument, but it too seems highly plausible. Assume that X is non-omnipotent, but is also the most powerful being in w1. In w2, X is less powerful than Y. If there is even a single possible world in which Y generates X, it follows that X is possibly generated. The same process can be used to show that any non-omnipotent being is possibly generated.
(8) seems indubitably true, at least on this particular use of "generated." If there is no time at which N can possibly not-exist, then N is just not the type of being that can be generated.
If each of these premises is correct, then (9) follows necessarily: a temporally necessary and omnipotent being exists.
It may be asked whether the argument requires that the past be infinite, but I don't think that's the case. Far from assuming the infinity of the past, (3) only requires that there be no time at which the statement, "something exists," is false. (3) is correct whether the past is finite or infinite. I take it that the temporally necessary being, granting that the past is finite, existed in an undifferentiated time at the beginning.