Page:Mind (New Series) Volume 15.djvu/519

SYMBOLIC REASONING. 505- half the infinity oo 3 . Similarly an infinity oo '" may be either infinite or infinitesimal in comparison with another in- finity oo ". 2. But, it may be asked, what is it exactly that separates the infinite from the finite ? Where is the exact line of demarcation? Let F denote the class of finite positive numbers or ratios, made up of the individual ratios F a , F 2 , F 3> , etc. ; and let H denote the class of positive infinities, H p H 2 , H 3 , etc. Logical consistency requires that these two classes shall be considered mutually exclusive ; for it is clear that, speaking of any number or ratio A, the two statements A F and A H are mutually inconsistent. If any real ratio A }s finite, it cannot be infinite ; and if it is infinite it cannot be finite. Neither can it, however large, be on the borderland between the two classes. These statements may be ex- pressed by a single formula, (A F A H ) 1 '. 3. For example, let M denote a million. The number M M (which means M x M x M x . . . etc., up to a million factors) is inconceivably large so large that the volume of the earth (or even of the sun, or of a sphere enclosing our whole solar system) divided by that of the smallest drop of water, would be an exceedingly small number in comparison ; yet the number M M belongs to the finite class F and not to the infinite class H. So does the number M MM , which is inconceivably large even in comparison with the inconceiv- ably large number M M ; and so does M MMM , which is incon- ceivably large even in comparison with M MM . And we might carry the ascending comparison further, till the hand got weary of the repetition of exponents, without finding any number or ratio that belongs to the infinite class H, or that does not belong to the finite class F. Are the finite F and the infinite H then both indefinable? Is there no quality Q which we can assert of every F and deny of every H ? There is, and it is this : Every finite number or ratio F, however large, is expressible, either exactly or (like TT) approximately, in the decimal or some other conventional notation, in terms of some finite number or ratio (such as 10 or 100 or 1,000,000, etc.) ; whereas no infinite number or ratio is, either exactly or approxi- mately, expressible solely in terms of any finite number or numbers. Thus, M M , M MM , M MMM , etc., though inconceivably large, are all finite, because they are all expressible in terms of the finite and known number M ; whereas H 1? 211^ T VHj, and generally FH (whatever be the finite number F and the infinite number H) are all infinities, because they are too large to be expressible, either exactly or approximately, solely in terms of any known numbers or ratios however large.