 Richard Dedekind (Dedekind 1887) had also listed the properties of numbers with similar looking axioms, using the notion of chain, an infinite sequence of sets, each a subset of the next, that is well ordered and has the structure of the natural numbers. Dedekind then proves that the principle of induction (Axiom 5 above) holds for chains. (See entry on  Dedekind).  Although Russell finds it “most remarkable that Dedekind’s previous assumptions suffice to demonstrate this theorem” (Russell 1903, §236), he compares the two approaches, of Peano and Dedekind, with respect to simplicity and their differing ways of treating mathematical induction, and concludes that: