boundary of X (being considered a part of a larger entity) is defined; but in practice, as a rule, the boundary of X (which may have no boundary) does not concern us, and all we need is the boundary of some enclosed part of it.
Connectedness and Continuity.—16. This takes us directly to the very important concept of continuity, which the concept of boundary enables us to define. We shall say that an entity is connected, if any and every one of its possible dissections gives us two joined entities; then by the continuity of a given entity we understand the fact that every one of its possible divisions (division meaning repeated dissection) gives us an aggregate of entities such that no matter which member or collection of members obtained by the division be taken (save the collection which is the complete entity), such member or collection will always be joined to at least one other member of the aggregate. From this it follows that given any two separate, connected parts of a continuous aggregate X, say A and B, it is always possible to find another connected part of X, say C, which will form a connected entity with A and B (is either joined to both or intersects both). A continuous aggregate of elements is called a continuum.
Ordinal Characteristic.—17. In an earlier paragraph we found that a given entity possessed extension by virtue of a certain class of attributes, which we denoted by the term extensional characteristic; similarly in the case of ordered aggregates we must assume the existence of something with respect to which the aggregate is ordered—by virtue of which the elements of the aggregate are arranged in one particular way and in no other. Without making any assumption as to the nature of this something, which we may, analogously to the extensional case, call the ordinal characteristic (or the ordering relation), we can readily see that this characteristic must in a certain way be the same for all elements of the aggregate, and in a certain way different for every one of the elements: the sameness of the characteristic for a given aggregate will depend upon the nature of this aggregate; the difference for the various elements will be either a result of convention, of a rule which we arbitrarily set up for the case in question, or—as is the case when the aggregate is given us already ordered—a result of comparison of the various elements, and consequent determination of the characteristic in which they all differ. Usually the comparison consists in measuring the amount of a certain quality possessed by the various elements; and we say that the various elements differ as to the “degree” or
