
Numbers of Number- Classes in General. 299 
where vy is any finite ordinal number*. Proceeding in this 
way, we establish a one-one correspondence between the 
double series (w,g), where 
at+6<v 
(w being some definite number of the second class), and the 
series (a,), where 
y<u 
For example, in the above first two stages of the enumeration, 
when a and @ are either both finite or such that 
l<a<o, oXP<wo. 2, 
or 
ax<a<@.2, 1<B<ao, 
it is easily seen to be necessary and sufficient that 
a+tB<o. 2. 
The general determination of the terms of (uz, g) which 
have been enumerated at any stage is, then, exactly analogous 
to that determination when each row and each column is of 
type w. In the latter case f, at any of the stages marked by 
the complete enumeration of a diagonal, the terms of (w,,¢) 
enumerated are those for which 
atB<y. 
where v is some finite ordinal number. 
Every term of the whole series (u,,s) is evidently such 
there is some number « of the first or second number-class 
such that 
atB<e; 
and consequently the correspondent of every term w,,g is 
some term a,, where y is a definite number of the first or 
second number-class. 
* In the actual construction of a relation like (5), we may proceed as 
follows :—To (u, ,), where l<a<o, oXS<w. 2, let the simple series 
(a,) correspond, by letting the term a,, 4, correspond to the term u, 
—_ A=p—14}(utv)(p+y—1), 
A, #, and y being finite ordinal numbers. Again, let a. 4, correspond 
t0 Uysy » Of (uw, g) When o<a<o.2, 1<8<a, where 
A=p+3(u+v)(u+v—1). 
Finally, let a’, ,,=4,,,,(A=0, 2, 4,...), 4, = 44, (A=L, 3, 5, ...). 
+ Cf. my paper already quoted, p. 4 and note. 
,W+v 
° 
