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163-165] Normally ordered aggregates 
points of their end-points does form an aggregate of order-type 6, when the 
elements, consisting partly of points, and partly of intervals, are taken in the 
order in which they occur in the continuum. 
164. The order-type *co + o> may be denoted by tt, and is the order-type 
of the negative and positive integers in their natural order. This order-type 
has properties distinct from those of co. For example, n +a> has been shewn 
to be identical with w, where n is a finite integer; but n + tt is not identical 
with tr. From either of the equations n + tt = m + tt, or tt + n = tt 4- m, 
there follows m —n, or more generally f: 
If n, n' are finite integers, £ and £' other order-types, from the equation 
n + tt + £ = n' + tt + £', there follows n — ri, £ ?= 
To prove this theorem, we observe that, if the two aggregates be placed 
into similar correspondence, the lowest elements correspond to one another, 
then the second, and so on; hence n = ri is proved at once: and we now have 
7T + £ = 7T + £". 
When two simply ordered aggregates M„ + Z, N n -f Z', of order-types 
tt + £, tt + £are placed in correspondence in order, either M„ corresponds 
to or corresponds to a part of N n , or else N, r corresponds to a part 
ot M n . In the last two cases the order-type tt must be split up into 
tt = tt x + tt 2 , where tt = 7r 1} and tt 2 is some other order-type; but from the 
definition tt = *o) + to, it is clear that every mode of dividing tt into two 
parts, without altering the relative order of the elements, leaves it in the 
form *co + co; hence it is impossible that tt = ttj + tt 2 , and tt = tt 1 ; and there 
fore M„ corresponds to . Hence also Z corresponds to Z'; or £= 
NORMALLY ORDERED AGGREGATES. 
165. The order-type of a simply ordered aggregate is, as we have already 
seen, such that the structure of the aggregate, as revealed by an examination 
of the sequences contained in it, may be of the most varied character; the 
various sequences may be ascending or descending ones, and may, or may not, 
have a limiting element within the aggregate. 
Of all the possible order-types, those are of especial importance which 
have been defined by Cantor as the order-types of normally ordered aggregates 
(wohlgeordnete Mengen). 
A normally ordered aggregate M is one which satisfies the following 
conditions: 
(1) M has an element m, of lower rank than all the other elements. 
(2) If Af is any part of M, and if M contains one or more elements 
which are of higher rank than all the elements of M 1} then there exists one 
f Bernstein, loc. cit., p. 9. 
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