Causal well-foundedness

I have defended causal finitism at length, mainly (but not only) as a way of getting out of paradoxes.

I want to think about a weaker thesis than causal finitism today. Causal finitism says that there is no item that has infinitely many things causally prior to it.

A weaker thesis is that causal priority (which I will assume to be transitive) is causal well-foundedness, namely that any set S of items contains a minimal item—an item that has nothing in S causally prior to it. Assuming the Axiom of Choice (and its applicability to sets of “items”), this is equivalent to saying there are no backwards infinite regresses ... < A₋₃ < A₋₂ < A₋₁ < A₀, where < indicates causal priority.

The arrangement of the Grim Reapers appears not to be well-founded: each one’s actions respond to the actions of the one before. So causal well-foundedness seems to rule out that paradox just as well as causal finitism.

At the same time, just as causal finitism implies the existence of a first cause (and hence yields a kalaam argument), so does causal well-foundedness: just take the set of all items that are causally prior to something, and a minimal item will be a first cause.

So far, no differences.

But alas once we start looking at other paradoxes, differences arise. Thomson’s lamp is unaffected by causal well-foundedness: the story is well-founded. Similarly, it seems one can shuffle an infinite deck of cards using a well-founded supertask. Likewise, there are paradoxes arising from from infinitely many things affecting a single result without any chain, like the paradox where infinitely many people roll independent and fair dice without seeing the results, and an angel announces that all but finitely many got sixes.

So you get what you pay for. Causal well-foundedness is a weaker assumption than causal finitism, but it also rules out fewer paradoxes.