(3) Let "0" mean the number one, let "number" mean the set and let "successor" mean "half." Then all Peano's five axioms will be true of this set. It is clear that such examples might be multiplied indefinitely. In fact, given any series [Pg 7] which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms. This is easily seen, though the formal proof is somewhat long. Let "0" mean , let "number" mean the whole set of terms, and let the "successor" of mean . Then (1) "0 is a number," i.e. is a member of the set. (2) "The successor of any number is a number," i.e. taking any term in the set, is also in the set. (3) "No two numbers have the same successor," i.e. if and are two different members of the set, and are different; this results from the fact that (by hypothesis) there are no repetitions in the set. (4) "0 is not the successor of any number," i.e. no term in the set comes before . (5) This becomes: Any property which belongs to , and belongs to provided it belongs to , belongs to all the 's. This follows from the corresponding property for numbers. A series of the form in which there is a first term, a successor to each term (so that there is no last term), no repetitions, and every term can be reached from the start in a finite number of steps, is called a progression. Progressions are of great importance in the principles of mathematics. As we have just seen, every progression verifies Peano's five axioms. It can be proved, conversely, that every series which verifies Peano's five axioms is a progression. Hence these five axioms may be used to define the class of progressions: "progressions" are "those series which verify these five axioms." Any progression may be taken as the basis of pure mathematics: we may give the name "0" to its first term, the name "number" to the whole set of its terms, and the name "successor" to the next in the progression. The progression need not be composed of numbers: it may be [Pg 8] composed of points in space, or moments of time, or any other terms of which there is an infinite supply. Each different progression will give rise to a different interpretation of all the propositions of traditional pure mathematics; all these possible interpretations will be equally true. In Peano's system there is nothing to enable us to distinguish between these different interpretations of his primitive ideas. It is assumed that we know what is meant by "0," and that we shall not suppose that this symbol means 100 or Cleopatra's Needle or any of the other things that it might mean.