Continuity of time and causation

In a standard causal deterministic system, given three times t₁ < t₂ < t₃, the state of the system at t₁ causes the state of the system at t₃ by means of the state of the state of system at t₂. If time is infinitely subdivided, then the state of the system at t₂ causes the state of the system at t₃ by means of the state of the system at t_2.5 (where t₂ < t_2.5 < t₃), and so on. This is an infinite regress. And it’s vicious, because it’s a dependency regress.

Here is one way to see that it’s a dependency regress. Imagine a really unpleasant situation where you need to kill Hitler, but the only way to kill Hitler is to initiate a continuous causal chain that proceeds through Hitler’s uncountably many henchmen, set up so that at every time strictly between t₁ and t₂ a henchman dies, and their death is caused by the death of each previously dead henchman; at t₁, Himmler is directly shot by you, and at t₂, Hitler dies because of the previous henchman deaths. It is clear that in this case every henchman’s death is intended as a means to Hitler’s death. This matters morally. If it turns out that any person in the causal chain is actually innocent, then the Principle of Double Effect will not allow you to kill Hitler by killing Himmler. For the causal chain from Himmler’s death through to Hitler’s proceeds by means of that innocent. But an outcome depends on its means, so a regress of means is a dependency regress.

If there are no vicious infinite regresses, it follows that one cannot have deterministic causal chains intimately tied to infinitely subdivided time. In fact, I think nothing hangs on determinism here.