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 ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.