Transfinite Numbers. 449 



to the class of which w^ is the first term. Then 

 M/3+1 =tf /3 + #, 



if .e < X^ + i then .I'T^, 



in that case ^+1 — &*/3 4-# = tt~ 



which is impossible. 



•'• * = ty+l. 



8. At this stage it is desirable to state clearly the con- 

 ditions necessary for proving by induction that a proposition 

 is true for every ordinal (or every transfinite ordinal). If a 

 proposition (or set of propositions) k is true for every ordinal 

 belonging to some initial segment of the ordinals (or of the 

 transfinite ordinals), and if it be proved that if /c is true for 

 every ordinal (or every transfinite ordinal) less than ft, then 

 it is also true for ft, then it follows that k is true for every 

 ordinal (or for every transfinite ordinal). It should be observed 

 that this statement includes both the case where ft has a term 

 next before it, and the case where ft has no term next 

 before it. 



It is not sufficient to prove that if k is true for ft and the 

 ordinals less than ft, then it is true for ft + 1. 



9. I now proceed to define (o@ where ft is any ordinal : — 



co 2 means coco. 



co v+1 means g>"g> (y any finite ordinal). 



Therefore the meaning of co v is determinate where v is any 

 finite ordinal, co^ (ft transfinite) is defined as follows : — 



In the first place if ft denotes a type of series with no last 

 term (i. e. if ft has no immediate predecessor), then o>0 de- 

 notes the first ordinal after all the ordinals aft (y</3). 



If ft = y + 1 then co? means a> Y a). 



It is clear from this definition that &/ denotes an ordinal 

 uniquely determined, provided that co y denotes an ordinal 

 uniquely determined, for every value of y less than ft. But 

 o)^ does denote an ordinal uniquely determined when ft is 

 finite. 



.'. co? denotes an ordinal uniquely determined for every 

 value of ft. 



I proceed to prove that co a+ ^ = co a oo^. 



This follows at once from the definition when ft is finite. 

 Phil. Mag. S. 6. Vol. 10. No. 5$. Oct. 1805. 2 I 



