Hume asks the question of how a finite effect can have an infinite cause. Kant says that you can base it on the moral argument. If you are explaining the subsets, you have the explanation of the infinite set (according to Hume). However, in explaining the first member of a finite set, one must go out of that set. Focus on a series in which a member’s existence is explained by the preceding cause. In explaining a set you go outside of the set. The point is not if it’s an infinite set but if it gets outside of the set (relies on PSRb in a way–that there must be some sufficient reasons for any positive fact and denies brute facts.)
Hume also argues against the CA on a priori grounds. He formulates the CA in a way that combines causal (Thomistic or kalam) and reason (Leibnizian). He considers the causal closure of the universe. Under his idea of how the universe is he considers it to be a causally closed system, which, by definition, rules out any external causation. Any a priori argument is absurd to Hume—it’s impossible to deduce the existence of God from fixed necessary premises or Kantian/Newtonian spacetime intuitions.
The biggest problem with Hume’s criticisms is that he believes the argument is primarily a priori. The argument is based on synthetic a posteriori arguments, which he should accept as being valid. By experience it seems to be the case that everything in the universe is contingent or has a cause to it. Hume has an interesting way of setting up the CA and his argument against natural theology. He uses fixed premises in the problematic (quaestio) form of inquiry. I think an interrogative approach is more appropriate in this sense. Inquiring to the states of affairs and seeking an explanation without any prior conception of what the truth may be [as with deduction]. There were no inductive or abductive methods applied in this way. He deduced his explanation from observations produced by his rationalistic positivism. Thus, following Kant’s influence on Hume, the sensorium is transferred to the person and the intellect does not draw its conclusions from nature but it imposes its conclusions upon nature. Hume did not allow for invention, free invention, or discovery. Hence, Hume’s model of inquiry is already suspect and problematic in his methodology.