 Considered more generally, what Dedekind has introduced is the natural numbers conceived of as finite “ordinal” numbers (or counting numbers: the first, the second, etc.). Later he adds an explanation of how their usual employment as finite “cardinal” numbers (answering to the question: how many?) can be recovered. This is done by using initial segments of the number series as tallies: for any set we can ask which such segment, if any, can be mapped one-to-one onto it, thus measuring its “cardinality”. (A set turns out to be finite in the sense defined above if and only if there exists such an initial segment of the natural numbers series.) Dedekind rounds off his essay by showing how several basic and so far merely assumed facts in arithmetic can now be proved as well. Especially significant is his purely “logical” justification of proofs by mathematical induction and definitions by recursion (based on his theory of chains).