Since we know that there is at least one contingent fact, we can identify C with the cosmos, and use Theorem 1 to conclude that the cosmos has a cause that is a necessary fact, a First Cause.
We know that all (or nearly all) wholly contingent facts have causes that the world is such a wholly contingent fact, and therefore we may conclude that the world has a cause, unless some relevant consideration pointing to the opposite conclusion can be produced.
Axiom 7 implies that each wholly contingent fact is caused: it does not imply that every correlation can be explained.
The cosmological argument includes no such error: it is demonstrated that the cosmos is itself a wholly contingent fact and for that reason must have a cause.
Since the cosmos contains every wholly contingent fact as a part, no contingent fact can be more necessary (less contingent) than the cosmos.
C* is clearly a contingent fact, since if it were necessary, the cosmos itself would be necessary (by Axiom 5, veridicality).
However, the fact that the First Cause has caused a is itself a contingent fact, so the First Cause would have to cause the fact that it caused a and so on, ad infinitum.
Even though we have excellent empirical evidence for the generalization that wholly contingent facts have causes, it is hard to see how any amount of data could settle conclusively the question of whether or not this generalization (Axiom 7) admits of exceptions.
First, it is hard to see why the abundant success of empirical science in finding causes for contingent facts does not provide overwhelming empirical support for the generalization to all contingent facts.
If we deny that there are any contingent facts, then we must conclude that we live in a world in which all three modalities -- possibility, actuality, and necessity -- collapse together.
Even if there are infinite regresses of causes within the totality of contingent facts, the totality itself must have a cause that is outside it and, hence, a cause that is necessary.