Even if life will eventually die out, we can pretend G is infinite, as a simplifying assumption, similarly to how physicists and chemists pretend crystals fill up infinite space in order to simplify talking about their translation symmetries. Under this simplifying assumption, "finite" and "infinite" can serve as concrete alternatives to vague slippery predicates like "small" and "large".
The following quote from Origin of Species reinforces that Darwin definitely didn't take for granted that life will someday end: "Nevertheless so profound is our ignorance, and so high our presumption, that we marvel when we hear of the extinction of an organic being; and as we do not see the cause, we invoke cataclysms to desolate the world, or invent laws on the duration of the forms of life!"
I'm out of my depths here, but isn't this trivially true regardless of means of reproduction?
You can't be your own parent so there's no directed cycles, and without directed cycles you can't have infinite directed paths in a finite graph.
Maybe he meant there's no path that starts from a root and ends in the leaf that has these properties? The leaf being the first life on earth in this example.
I know the tree of life isn't a tree, but it surely is acyclic, right?