PRIZELL: FOUNDATIONS OF ARITHMETIC. 399 



simply ordered according to the higher rule. X is itself 

 an unbounded set and contains a sub set holoedrically 

 isomorphic with R as regards each rule of combination. 

 Thus the whole set of absolute numbers is deduced 

 from the set of absolute rational numbers by using only 

 principles of order. 



76. The preceding development does not take ac- 

 count of the possibility that K[f„] may not really di- 

 vide S. There may be no symbol which follows every 

 ti. Then Wi is an empty class and Vi is taken to coin- 

 cide with S. Thus if K[f„l and KLg„] are both 

 series of this kind, they both have the same W and V, 

 whence if new symbols f ^ , g„ were to be introduced 

 the principle of definition used in § 75 would lead us to 

 declare f^ = g^,. Therefore we assign to the totality of 

 the sets K [ f „ J for which Wi is an empty class a single 

 symbol Z which is to follow every element of X. Then 

 following § 75, Z o g^ is to be defined as h^ where h„ = 

 f„ogn and f« = Z. But for K[h^], Wi is empty, 

 therefore Z oic = Zog^ = h^ = Z. That is, Z-{-x = 

 Z=^x^Z,Zx = Z = xZ, and similarly Z + Z = Z = 

 ZZ = Z^ = Z^. The associative, commutative and dis- 

 tributive laws still hold as well as the first group prop- 

 erty, but the semigroup is destroyed for both rules by 

 violating the law of equals with unequals. 



77. It is also possible that no symbol in S precedes 

 every f^. Then Vj is an empty class and Wg coincides 

 with S. On the same principle as in § 76 we assign to 

 the totality of series of this kind a single symbol v to 

 precede every x and define v o x = h^ where h„ = f„ o g„, 

 g«, = cc, f^ = V. And now the two rules must be dis- 

 tinguished. Since no x precedes every f^ the sets 

 K[f„ + g„] and K[g„J separate S into the same V 

 and W. Therefore v-|-aj = h^ = g^ = ic + v. And for 

 the same reason no s can precede every f„ g„ 



.'. va; = (fg)^==v = xv 



