Transfinite Numbers. 451 



it follows from our premisses that the cardinal number of the 

 aggregate of terms in each segment is tf , for if y be finite 

 C!(« y ) = Kp, and if 7 be transfinite C{co y ) = C(y) = X . 



The series of segments is a series of type ft, and C(ft) =tto- 



Therefore the cardinal number of the whole ao-oreo-ate of 

 terms is ^J = K . 



Therefore co? also belongs to the 2nd class. 



Now it is easy to see that co co belongs to the 2nd class *> 

 for the series of ordinals less than &> w is 



1,2,3,... 



ft), CO + 1 , ... co2, co2 + 1, ... ft>3, . . . COV, . . . 



ft) 2 , ft) 2 -f-l, eo 2 2, . . . co*v, 



ft) 3 , a) 3 + 1, 



ft) 1 



i. e. it is a progression of segments each of type co v (for some 

 finite value of v). 



The cardinal number of the aggregate of terms in each 

 segment is tt . 



,-. C(o>») = ^ = K , 



... c(ft)"+") = K . 



Therefore, from what I have proved above, it follows that 

 if ft be any ordinal of the 2nd class, then C(o)^) = C(/3) ; or, 

 in other words, co? also belongs to the 2nd class. 



11. I now proceed to prove that tf^tf^ for all values of 

 ft (finite and transfinite). 



Consider the two propositions 



0(O = C(/3)-) 

 {0(/3)} 3 =C(/3)/ W 



It has been proved that these two propositions are true if 

 ft be any ordinal of the second class. 



I shall prove that if they are true for all transfinite ordinals 

 less than ft then they are also true for ft. 



Assuming the propositions A to be true for all transfinite 

 ordinals < ft. 



* This can also be verified by arranging the finite numbers in a series 

 of type co w , viz., first the primes in order of magnitude, then the numbers 

 with two factors in a series of type w 2 , then the numbers with three 

 factors in a series of type w 3 , and so on. 



2 1 2 



