# The Principles of Mathematics (1903)

## § 121

The mutual independence of these five propositions has been demonstrated by Peano and Padoa as follows[88]. (1) Giving the usual meanings to 0 and successor, but denoting by number finite integers other than 0, all the above propositions except the first are true. (2) Giving the usual meanings to 0 and successor, but denoting by number only finite integers less than 10, or less than any other specified finite integer, all the above propositions are true except the second. (3) A series which begins by an antiperiod and then becomes periodic (for example, the digits in a decimal which becomes recurring after a certain number of places) will satisfy all the above propositions except the third. (4) A periodic series (such as the hours on the clock) satisfies all except the fourth of the primitive propositions. (5) Giving to successor the meaning greater by 2, so that the successor of 0 is 2, and of 2 is 4, and so on, all the primitive propositions are satisfied except the fifth, which is not satisfied if s be the class of even numbers including 0. Thus no one of the five primitive propositions can be deduced from the other four.(§ 121 ¶ 1)

§ 121 n. 1. F. 1899, p. 30