FRIZELL: FOUNDATIONS OF ARITHMETIC. 385 



nation eoa where a denotes a previously defined mem- 

 ber of the set is also well ordered, infinite and forms a 

 K-class with respect to the given rule. Proof : By 

 hypothesis the set is to contain eoe = e', eoe' = e", 

 eoe" = e'", . . . Since it is to be an ordered set no two 

 of its members can be equal .".to every element of the 

 whole set can be assigned some one of the subset e', 

 e'", e", . . . That is, the set is infinite. And byb) 

 every element has an immediate successor. Finally we 

 have a test for equality which satisfies the require- 

 ments of § 1. 



9. Scholium. A set defined as in § 8 contains no 

 modulus. In what follows it will be referred to as an 

 e-sei. 



10. PosUdates. We postulate an e-set for a lower 

 rule of combination which shall contain a lower symbol 

 u and e-sets for both this and a higher rule which shall 

 both contain a higher symbol w, the definitions of the 

 rules of combination to be completed in §§ 17, 26, 

 28, 30. 



11. The different sets K[uo], K[wo], K[^6'^] are 

 to be so ordered that the lower shall precede the higher. 

 Thus they form together a well ordered set 



u, uou = u', uou' = u", uou" = n"', .... 



w, wow = wu', wowu' = wu", wowu" = wu'", . . . 



wntv = w''\ wnw''' = w''", waW" ^W'", . . . 



12. Postulates. We postulate 6-sets for the lower 

 rule which shall contain respectively each of the sym- 

 bols w' , w" , iv^" , ... in succession. Thus 

 corresponding to every a of the set K[i^n] we shall 

 have an e-set K[ao]:a, aoa ^ au' , aoau' = au" , 

 . . . Now taking e. g. a = w'" , by the principle of 

 § 11 every member of K[ii'""'o] precedes ww'" = w^'' 

 and therefore falls between iv'"' and w^\ That is, 

 all these new e-sets are interpolated between successive 



