Arguing for a first cause without assuming transitivity of causation

This is another technical research note.

Here is a cosmological argument that does not assume causation is transitive.

Suppose the set of causes forms a directed acyclic graph where the edges are cases of causation. Say that x weakly causes y provided that there is a chain of causation from x to y.

Suppose that any internal chain C of causes that is reverse well-ordered under weak causation is founded, i.e., there is a weak cause c of everything in the chain except c itself. Assume there are no causal cycles. Suppose the following technical condition:

• There is no sequence of causes considered with respect to weak causation that is reverse-order isomorphic to ω₁.

Finally assume there is at least one cause. It follows (assuming the Axiom of Choice) that there is an uncaused cause.

The technical assumption seems pretty plausible. Nobody to my knowledge has proposed a model of the universe where the causal history contains a reverse copy of ω₁. It would be like thinking the past has not only an infinite but an uncountable number of days. (An uncountable number of past moments is tame—though of course it violates causal finitism—but an uncountable number of past days would be wild.)

Sketch of proof: Let G be the directed acyclic graph whose vertices are causes and where (a,b) is an edge iff b causes a. Let R be the partial order generated by the edge directions. Our technical assumption says that ω₁ does not order embed into G. Let C be any well-ordered R-chain of vertices in G. If we can show in general that C has an R-upper bound, by a standard refinement of Zorn’s Lemma it will follow that G has an R-maximal element, and that’s an caused cause.

Since ω₁ does not embed in G, C is countable. Passing to a cofinal (with respect to R) subset of C that is isomorphic to ω, we can assume C has order type ω. Now insert finite paths between all pairs of successive vertices of C (this uses AC, and the acyclicity implies that the paths all run in the same direction as the order on C). The resulting set C′ of vertices is an internal chain with respect to graph direction, and so it has an R-upper bound by the foundedness assumption on chains.