 What it means to be simply infinite can now be captured in four conditions: Consider a set S and a subset N of S (possibly equal to S). N is called simply infinite if there exists a function f on S and an element 1 of N such that: (i) f maps N into itself; (ii) N is the chain (minimal closure) of \(\{1\}\) in S under \(\,f\); (iii) 1 is not in the image of N under \(f\); and (iv) \(f\) is one-to-one. While at first unfamiliar, it is not hard to see that these Dedekindian conditions are a notational variant of Peano’s well-known axioms for the natural numbers. In particular, condition (ii) is a version of the axiom of mathematical induction. These axioms are thus properly called the Dedekind-Peano axioms. (Peano, who published his corresponding work in 1889, acknowledged Dedekind’s priority; cf. Ferreirós 2005.) As is also not hard to see, any simple infinity will consist of a first element corresponding to 1, a second element corresponding to \(f(1)\), a third corresponding to \(f(f(1))\), and so on, just like any model of the Dedekind-Peano axioms.