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The principal one amongst all the traditional objections to the Infinite is, as we have seen, the thought that the Infinite, as such, is merely an endlessly sought or an endlessly incomplete somewhat; while the real, as such, is very rightly to be viewed as the determinate. Hence, the actually Infinite, one insists, would be at once determinate and indeterminate, and so would be contradictory.

Now, whatever may be said about the actually Infinite, we have already seen that the infinite of the merely conceptual but valid type, the infinite of the realm of mathematical possibilities, is certainly as determinate a conception as any merely universal idea can ever be, and, as thus determinate, involves no contradiction whatever. Cling to our Third Conception of Reality; and then, indeed, there can be no doubt whatever that the Infinite is real. For there is no contradiction, there is only a necessarily valid truth involved in saying that to any whole number r, however large, there inevitably does correspond one number, and only one, which stands amongst all numbers as the rth member in the ordered series of whole numbers that are squares, or in the ordered series of the cubes, or in the ordered series, if you please, of numbers of the form of ɑ100 or ɑ1000, where the exponent is fixed, but where the number that is to be raised to the power indicated takes successively the series of values, 1, 2, 3,. . . r. The inevitable result is that to every whole number r, without a possible exception, there corresponds, in the realm of validity, and corresponds uniquely, just that particular whole number which you get if you raise r to the second, third, or hundredth, or thousandth power. Moreover, this ideal ordering of all the whole numbers, without exception, in a one-to-one relation (let us say) to their own thousandth powers, is in such wise predetermined by the very nature of number that, if you undertake to calculate the thousandth power (let us say) of the number 80,000,000, your result is in no wise left to you, as a bare possibility that your private will can capriciously decide.