The following example is even more surprising. Write the natural numbers 1, 2, 3, 4, ... in the top row, and the even numbers 2, 4, 6, 8, ... in the bottom row, so that under each number in the top row stands its double in the bottom row. Then, as before, the number of numbers in the two rows is the same, yet the second row results from taking away all the odd numbers--an infinite collection--from the top row. This example is given by Leibniz to prove that there can be no infinite numbers. He believed in infinite collections, but, since he thought that a number must always be increased when it is added to and diminished when it is subtracted from, he maintained that infinite collections do not have numbers. "The number of all numbers," he says, "implies a contradiction, which I show thus: To any number there is a corresponding number equal to its double. Therefore the number of all numbers is not greater than the number of even numbers, _i.e._ the whole is not greater than its part."[49] In dealing with this argument, we ought to substitute "the number of all finite numbers" for "the number of all numbers"; we then obtain exactly the illustration given by our two rows, one containing all the finite numbers, the other only the even finite numbers. It will be seen that Leibniz regards it as self-contradictory to maintain that the whole is not greater than its part. But the word "greater" is one which is capable of many meanings; for our purpose, we must substitute the less ambiguous phrase "containing a greater number of terms." In this sense, it is not self-contradictory for whole and part to be equal; it is the realisation of this fact which has made the modern theory of infinity possible. [49] _Phil. Werke_, Gerhardt's edition, vol. i. p. 338. There is an interesting discussion of the reflexiveness of infinite wholes in the first of Galileo's Dialogues on Motion. I quote from a translation published in 1730.[50] The personages in the dialogue are Salviati, Sagredo, and Simplicius, and they reason as follows: "_Simp._ Here already arises a Doubt which I think is not to be resolv'd; and that is this: Since 'tis plain that one Line is given greater than another, and since both contain infinite Points, we must surely necessarily infer, that we have found in the same Species something greater than Infinite, since the Infinity of Points of the greater Line exceeds the Infinity of Points of the lesser. But now, to assign an Infinite greater than an Infinite, is what I can't possibly conceive.