1 1 1 



406 KANSAS UNIVERSITY SCIENCE BULLETIN. 



member of this series by adding a second quotient 

 results again type w . Therefore the fractions of the 

 form -^ ^- may be arranged in series ordinally sim- 

 ilar to 1, 2, . . . w-\-l, . . . 2w, . . . Sw, . . . 

 i. e. of type w^ . Annexing a third quotient replaces 

 each element of this set by a series of type w so that 

 we include types w^-\-l, w^-\-2, . . . 2w^ 



Sw^, . . . i. e. the class of fractions ^ ^ „ ^ ^ 



' qi+ q2+ q-i 



is ordered in type w^. 



Therefore the whole set of simple continued frac- 

 tions, comprehending types ^(; , . . . ^% . . . w% . . . 

 w^, . . . constitutes, as ordered, a series of type 



93. Now it is easy to arrange the natural numbers 

 also in a series of type w ' as follows. We know that 

 the class of prime numbers can be put into one to one 

 correspondence with the whole set of natural numbers 

 (the number of primes is infinite). 



Therefore we can set up a one to one correspondence 

 ordinally between the prime numbers and the simple 

 continued fractions with a single quotient. By the same 

 reasoning the continued fractions with two quotients 

 are shown to be ordinally similar to the class of prod- 

 uct of two primes, i. e., if to every quotient q^ we 

 assign that prime pi for which q^ is the ordinal number 

 in sequence (so that 6. g. to the quotient 7 we assign 

 17) and likewise for a second prime pa and quotient q^, 

 and so on. 



Proceeding in this way the class of all products of N 

 primes is exhibited as ordinally similar to the class of 

 continued fractions -^ -^ * * * ^ • But the class of 

 all products of primes is the whole set of natural num- 

 bers. Therefore by § 92 the natural numbers may be 

 arranged in series w' . Q. E. D. 



94. In other words we have here a method whereby 



