THE NEW LOGICS. 165 



so called. M. Couturat begins by enunciating Peano's 

 five axioms, which are independent, as Signor Peano 

 and Signor Padoa have demonstrated, 



1. Zero is a whole number. 



2. Zero is not the sequent of any whole number. 



3. The sequent of a whole number is a whole 

 number. To which it would be well to add : every 

 whole number has a sequent. 



4. Two whole numbers are equal if their sequents 

 are equal. 



The 5th axiom is the principle of complete induction. 



M. Couturat considers these axioms as disguised 

 definitions ; they constitute the definition by postulates 

 of zero, of the " sequent," and of the whole number. 



But we have seen that, in order to allow of a 

 definition by postulates being accepted, we must be 

 able to establish that it implies no contradiction. 



Is this the case here ? Not in the very least. 



The demonstration cannot be made by example. 

 We cannot select a portion of whole numbers — for 

 instance, the three first — and demonstrate that they 

 satisfy the definition. 



If I take the series o, i, 2, I can readily see that 

 it satisfies axioms i, 2, 4, and 5 ; but in order that 

 it should satisfy axiom 3, it is further necessary that 

 3 should be a whole number, and consequently that 

 the series o, I, 2, 3 should satisfy the axioms. We 

 could verify that it satisfies axioms i, 2, 4, and 5, 

 but axiom 3 requires besides that 4 should be a 

 whole number, and that the series o, i, 2, 3, 4 should 

 satisfy the axioms, and so on indefinitely. 



It is, therefore, impossible to demonstrate the 

 axioms for some whole numbers without demonstrat- 



