HHHii 
310 
ORDINAL NUMBERS 
Let us begin by ordering the sections of 21 and 23 as in 272, 3. 
Let B denote the aggregate of sections of 23 to which similar sec 
tions of 2t do not correspond. Then B is well ordered and has a 
first section, say Sb. Let /3 < b. Then to S/3 in 23 corresponds 
by hypothesis a similar section Sa in 2f. On the other hand, to 
any section Sa' of 21 corresponds a similar section Sb' of 23. Ob 
viously b' < b. Thus to any section of 21 corresponds a similar 
section of Sb and conversely. Hence 2l=**S7> by 277, 1. 
279. Let 21, 23 be well ordered. Either 21 is similar to 23 or one 
is similar to a section of the other. 
For either: 
1° To each section of 21 corresponds a similar section of 23 
and conversely; 
or 2° To each section of one corresponds a similar section of 
the other but not conversely ; 
or 3° There is at least one section in both 2i and 23 to which no 
similar section corresponds in the other. 
If 1° holds, 2t ^23 by 277, l. If 2° holds, either 21 or 23 is similar 
to a section of the other. 
We conclude by showing 3° is impossible. 
For let A be the set of sections of 21 to which no similar section 
in 23 corresponds. Let B have the same meaning for 23. If we 
suppose 21, 23 ordered as in 272, 3, A will have a first section say 
Sa, and B a first section S/3. 
Let a < a. Then to Sa in 21 corresponds by hypothesis a sec 
tion Sb of S/3 as in 278. Similarly if b' < /3, to Sb' of 23 corre 
sponds a section Sa' of Sa. Hut then Sa^S/3 by 277, l, and this 
contradicts the hypothesis. 
Ordinal Numbers 
280. 1. With each well ordered aggregate 21 we associate an 
attribute called its ordinal number, which we define as follows : 
1° If 2i — 23, they have the same ordinal number. 
2° If 2t — a section of 23, the ordinal number of 2f is less than 
that of 23.