(1) _Reflexiveness._--A number is said to be _reflexive_ when it is not increased by adding 1 to it. It follows at once that any finite number can be added to a reflexive number without increasing it. This property of infinite numbers was always thought, until recently, to be self-contradictory; but through the work of Georg Cantor it has come to be recognised that, though at first astonishing, it is no more self-contradictory than the fact that people at the antipodes do not tumble off. In virtue of this property, given any infinite collection of objects, any finite number of objects can be added or taken away without increasing or diminishing the number of the collection. Even an infinite number of objects may, under certain conditions, be added or taken away without altering the number. This may be made clearer by the help of some examples. Imagine all the natural numbers 0, 1, 2, 3, ... to be written down in a row, and immediately beneath them write down the numbers 1, 2, 3, 4, ..., so that 1 is under 0, 2 is under 1, and so on. Then every number in the top row has a number directly under it in the bottom row, and no number occurs twice in either row. It follows that the number of numbers in the two rows must be the same. But all the numbers that occur in the bottom row also occur in the top row, and one more, namely 0; thus the number of terms in the top row is obtained by adding one to the number of the bottom row. So long, therefore, as it was supposed that a number must be increased by adding 1 to it, this state of things constituted a contradiction, and led to the denial that there are infinite numbers. 0, 1, 2, 3, ... _n_ ... 1, 2, 3, 4, ... _n_ + 1 ...