Of these two papers, the second is a briefer but almost certainly later version of the first. The point of view underlying both papers is succinctly expressed by the opening sentence of 1909b:Anyone wishing to found arithmetic on the theory of natural numbers as the finite cardinal numbers, faces the task, first and foremost, of defining finite set; for cardinal number is by nature a property of sets, and every statement about finite cardinal numbers can always be expressed as one about finite sets.