A series is said to be "closed" (abgeschlossen) when every progression or regression contained in the series has a limit in the series. We then have the further definition:— A series is "perfect" when it is condensed in itself and closed, i.e. when every term is the limit of a progression or regression, and every progression or regression contained in the series has a limit in the series. In seeking a definition of continuity, what Cantor has in mind is the search for a definition which shall apply to the series of real numbers and to any series similar to that, but to no others. For this purpose we have to add a further property. Among the real numbers some are rational, some are irrational; although the number of irrationals is greater than the number of rationals, yet there are rationals between any two real numbers, however [Pg 103] little the two may differ. The number of rationals, as we saw, is . This gives a further property which suffices to characterise continuity completely, namely, the property of containing a class of 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.