But \aleph_{0}. "> In fact, as we shall see later, is a very important number, namely, the number of terms in a series which has "continuity" in the sense in which this word is used by Cantor. Assuming space and time to be continuous in this sense (as we commonly do in analytical geometry and kinematics), this will be the number of points in space or of instants in time; it will also be the number of points in any finite portion of space, whether [Pg 86] line, area, or volume. After , is the most important and interesting of infinite cardinal numbers. Although addition and multiplication are always possible with infinite cardinals, subtraction and division no longer give definite results, and cannot therefore be employed as they are employed in elementary arithmetic. Take subtraction to begin with: so long as the number subtracted is finite, all goes well; if the other number is reflexive, it remains unchanged. Thus , if is finite; so far, subtraction gives a perfectly definite result. But it is otherwise when we subtract from itself; we may then get any result, from 0 up to . This is easily seen by examples. From the inductive, numbers, take away the following collections of terms:— (1) All the inductive numbers—remainder, zero. (2) All the inductive numbers from onwards—remainder, the numbers from 0 to , numbering terms in all. (3) All the odd numbers—remainder, all the even numbers, numbering terms. All these are different ways of subtracting from , and all give different results. As regards division, very similar results follow from the fact that is unchanged when multiplied by 2 or 3 or any finite number or by . It follows that divided by may have any value from 1 up to . From the ambiguity of subtraction and division it results that negative numbers and ratios cannot be extended to infinite numbers. Addition, multiplication, and exponentiation proceed quite satisfactorily, but the inverse operations—subtraction, division, and extraction of roots—are ambiguous, and the notions that depend upon them fail when infinite numbers are concerned.