 Russell’s definition of natural numbers as classes of similar, or equinumerous, classes, first published in (Russell 1901), was his first logical construction, and was the model for those that followed. Similar classes are those that can be mapped one to one onto each other by some relation. The notion of a “one-to-one relation” is defined with logical notions: R is one-one when for every \(x\) there is a unique \(y\) such that \(x \rR y\), and for every such \(y\) in the range of \(\rR\) there is a unique such \(x\). These notions of existence and uniqueness come from logic, and so the notion of number is thus defined solely in terms of classes and of logical notions. Russell announced the goal of his logicist program in The Principles of Mathematics: “the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles…” (Russell 1903, xv). If  class  is also shown to be a logical notion, then this definition would complete the logicist program for the mathematics of natural numbers. 