Peano points out (loc. cit.) that other classes besides that of the finite integers satisfy the above five propositions. What he says is as follows: There is an infinity of systems satisfying all the primitive propositions. They are all verified, e.g., by replacing number and 0 by number other than 0 and 1. All the systems which satisfy the primitive propositions have a one-one correspondence with the numbers. Number is what is obtained from all these systems by abstraction; in other words, number is the system which has all the properties enunciated in the primitive propositions, and those only. This observation appears to me lacking in logical correctness. In the first place, the question arises: How are the various systems distinguished, which agree in satisfying the primitive propositions? How, for example, is the system beginning with 1 distinguished from that beginning with 0? To this question two different answers may be given. We may say that 0 and 1 are both primitive ideas, or at least that 0 is so, and that therefore 0 and 1 can be intrinsically distinguished, as yellow and blue are distinguished. But if we take this view--which, by the way, will have to be extended to the other primitive ideas, number and succession--we shall have to say that these three notions are what I call constants, and that there is no need of any such process of abstraction as Peano speaks of in the definition of number. In this method, 0, number, and succession appear, like other indefinables, as ideas which must be simply recognized. Their recognition yields what mathematicians call the existence-theorem, i.e. it assures us that there really are numbers. But this process leaves it doubtful whether numbers are logical constants or not, and therefore makes Arithmetic, according to the definition of Part I, Chapter I, primâ facie a branch of Applied Mathematics. Moreover it is evidently not the process which Peano has in mind. The other answer to the question consists in regarding 0, number, and succession as a class of three ideas belonging to a certain class of trios defined by the five primitive propositions. It is very easy so to state the matter that the five primitive propositions become transformed into the nominal definition of a certain class of trios. There are then no longer any indefinables or indemonstrables in our theory, which has become a pure piece of Logic. But 0, number and succession become variables, since they are only determined as one of the class of trios: moreover the existence-theorem now becomes doubtful, since we cannot know, except by the discovery of at least one actual trio of this class, that there are any such trios at all. One actual trio, however, would be a constant, and thus we require some method of giving constant values to 0, number, and succession. What we can show is that, if there is one such trio, there are an infinite number of them. For by striking out the first term from any class satisfying the conditions laid down concerning number, we always obtain a class which again satisfies the conditions in question. But even this statement, since the meaning of number is still in question, must be differently worded if circularity is to be avoided. Moreover we must ask ourselves: Is any process of abstraction from all systems satisfying the five axioms, such as Peano contemplates, logically possible? Every term of a class is the term it is, and satisfies some proposition which becomes false when another term of the class is substituted. There is therefore no term of the class which has merely the properties defining the class and no others. What Peano's process of abstraction really amounts to is the consideration of the class and variable members of it, to the exclusion of constant members. For only a variable member of the class will have only the properties by which the class is defined. Thus Peano does not succeed in indicating any constant meaning for 0, number, and succession, nor in showing that any constant meaning is possible, since the existence-theorem is not proved. His only method, therefore, is to say that at least one such constant meaning can be immediately perceived, but is not definable. This method is not logically unsound, but it is wholly different from the impossible abstraction which he suggests. And the proof of the mutual independence of his five propositions is only necessary in order to show that the definition of the class of trios determined by them is not redundant. Redundancy is not a logical error, but merely a defect of what may be called style. My object, in the above account of cardinal numbers, has been to prove, from general Logic, that there is one constant meaning which satisfies the above five propositions, and that this constant meaning should be called number, or rather finite cardinal number. And in this way, new indefinables and indemonstrables are wholly avoided; for when we have shown that the class of trios in question has at least one member, and when this member has been used to define number, we easily show that the class of trios has an infinite number of members, and we define the class by means of the five properties enumerated in Peano's primitive propositions. For the comprehension of the connection between Mathematics and Logic, this point is of very great importance, and similar points will occur constantly throughout the present work.(§ 122 ¶ 1)