Take any collection of things--we call that an _aggregate_. If an aggregate corresponds one to one with another aggregate, they are both said to have a number, and the same number. Subtract from the idea of an aggregate the idea of quality or kind, and order or arrangement, what is left is its _cardinal number_. If you subtract quality and not order, the result is an _ordinal number_. This reasoning applies both to finite and infinite aggregates; in fact, the Infinite may be said to possess most of the properties which we attach to the Finite. Two infinite aggregates, for example, can have an ordinal correspondence, and infinite aggregates submit, like finite ones, to arithmetical processes. The mathematician analyses still more closely the relation between the Finite and the Infinite, as follows:-- He starts from the aggregate, which he analyses into the Finite and the Infinite, and the latter he analyses into the Transfinite and the Absolute. Of these two elements, one only has till just lately been the subject of mathematical treatment--the one called the Transfinite. It is the transfinite subdivision of the Infinite to which the idea of number is applicable, and which is, therefore, in a sense inseparable from the Finite. Infinite numbers or series ought then to be more correctly described as _Transfinite_. But the processes of mathematics do not end here; they reach up to the idea of the _Absolute Infinite_, the conception of which has been attained in recent years by mathematical work. The results of this work may now be briefly summarised. I. The Absolute appears to have the same relation to the Transfinite as the Transfinite to the Finite. If the Finite deals with numbers, and the Transfinite with series of numbers, the Absolute deals with series of series. Thus there are at least two examples of the Infinite within our grasp which lead up to the idea of the Absolute. One is the class of all classes of propositions; the other is the series of all worlds of thought, in Dedekind’s sense. II. The Finite, Transfinite, and Absolute can be further defined in this way. There is no greatest finite number, but there is a least transfinite number, which has been called Aleph 0, and which can be proved to be greater than any possible finite number, however large, because if there were a last number it must be smaller than the sum of the whole series. There are unending series of Alephs or infinite numbers, which are as distinct from one another in idea as 1 is from 0, and which can no more be derived from one another by a mathematical process than 1 can be derived from 0, but can be reached in the same way by induction. Beyond the Transfinite we cannot discover in the Absolute the idea of least or of greatest.