The Meyer cosmological argument revisited

I’m playing with an alternative way to run the Meyer cosmological argument (with slightly weaker assumptions).

Suppose there is at least one cause, and there is a set of all causes.

Suppose causation is transitive. (If it’s not, replace “x causes y” with “there is a finite chain of causes running from x to y” throughout the argument.)

Define a causal circularity as a set of at least two causes such that for any distinct x and y in the set, x causes y and y causes x.

Define a causal chain as a set of causes such that for any distinct x and y in the chain, x causes y or y causes x but not both.

A causal chain S is reverse-well-ordered provided that any non-empty subset of it contains a unique effect, i.e., for any non-empty subset U of S, there is a z in U such that everything else in U is a cause of z.

The reverse well-order condition ensures, among other things, that each cause in the chain that is not the first cause in the chain has an immediate predecessor, and hence rules out “continuous chains”. This makes the chains more intuitively “chain-like”, and hence should make us feel more secure about applying our intuitions to them.

Say that a set S of causes is unfounded provided that there is no cause x that causes all the items in S perhaps other than itself.

Say that it is externally unfounded provided that there is no such cause outside the set S.

Then, given the Axiom of Choice, at least one of the following three is true:

1. There is an unfounded reverse-well-ordered causal chain.

2. There is an externally unfounded causal circularity.

3. There is an uncaused cause.

Hence we get an argument for an uncaused cause assuming all reverse-well-ordered causal chains are founded and any causal circularities are externally founded.

Proof: Suppose (1) and (2) are false. Write x ≤ y if x causes y or x = y. Write x ∼ y if x ≤ y and y ≤ x. For x a cause, let [x] be the equivalence class of x under ∼ in the set of all causes. Let C be the set of all equivalence classes of causes, and extend ≤ to C in the obvious way to provide a partial order on C.

Let U be a reverse-well-ordered ≤-chain in C. By the Axiom of Choice, let V be a set containing exactly one member from each ∼-equivalence class that is a member of U. This is a reverse-well-ordered causal chain. It must be founded by the falsity of (1). Thus, there is an x in V such that x ≤ y for all y in V. Thus [x] is a ≤-lower bound for U. By a slight generalization of Zorn’s Lemma requiring only well-ordered chains (this proof yields this generalization), there is a [u] in C that is ≤-minimal. If [u] is a singleton, then u is an uncaused cause. Suppose [u] has more than one member. Then [u] is a causal circularity, and hence has an external cause v by the falsity of (2). Then [v] ≤ [u] and [v] ≠ [u], contrary to [u] being ≤-minimal. Hence [u] cannot have more than one member, and we are done.