members in such a way that some of this class occur between any two terms of our series, however near together. This property, added to perfection, suffices to define a class of series which are all similar and are in fact a serial number. This class Cantor defines as that of continuous series.
We may slightly simplify his definition. To begin with, we say:
A "median class" of a series is a sub-class of the field such that members of it are to be found between any two terms of the series.
Thus the rationals are a median class in the series of real numbers. It is obvious that there cannot be median classes except in compact series.
We then find that Cantor's definition is equivalent to the following:—
A series is "continuous" when (1) it is Dedekindian, (2) it contains a median class having
terms.
To avoid confusion, we shall speak of this kind as "Cantorian continuity." It will be seen that it implies Dedekindian continuity, but the converse is not the case. All series having Cantorian continuity are similar, but not all series having Dedekindian continuity.