A causal finitist definition of the finite

Causal finitism says that nothing can have infinitely many causes. Interestingly, we can turn causal finitism around into a definition of the finite.

Say that a plurality of objects, the xs, is finite if and only if it possible for there to be a plurality of beings, the ys, such that (a) it is possible for the ys to have a common effect, and (b) it is possible for there to be a relation R such that whenever x₀ one of the xs, then there is exactly one of a y₀ among the xs such that Rx₀y₀.

Here's a way to make it plausible that the definition is extensionally correct if causal finitism is true. First, if the definition holds, then clearly there are no more of the xs than of the ys, and causal finitism together with (a) ensures that there are finitely many of the ys, so anything that the definition rules to be finite is indeed finite. Conversely, suppose the xs are a finite plurality. Then it should be possible for there to be a finite plurality of persons each of which thinks about a different one of the xs in such a way that each of the xs is thought about by one of the ys. Taking being thought about as the relation R makes the definition be satisfied.

Of course, on this account of finitude, causal finitism is trivial, for if a plurality of objects has an effect, then they satisfy the above definition if we take R to be identity. But what then becomes non-trivial is that our usual platitudes about the finite are correct.