Again, how do we know that the last number reached in the process of counting is the number of the objects counted? This is just one of those facts that are too familiar for their significance to be realised; but those who wish to be logicians must acquire the habit of dwelling upon such facts. There are two propositions involved in this fact: first, that the number of numbers from 1 up to any given number is that given number--for instance, the number of numbers from 1 to 100 is a hundred; secondly, that if a set of numbers can be used as names of a set of objects, each number occurring only once, then the number of numbers used as names is the same as the number of objects. The first of these propositions is capable of an easy arithmetical proof so long as finite numbers are concerned; but with infinite numbers, after the first, it ceases to be true. The second proposition remains true, and is in fact, as we shall see, an immediate consequence of the definition of number. But owing to the falsehood of the first proposition where infinite numbers are concerned, counting, even if it were practically possible, would not be a valid method of discovering the number of terms in an infinite collection, and would in fact give different results according to the manner in which it was carried out.
There are two respects in which the infinite numbers that are known differ from finite numbers: first, infinite numbers have, while finite numbers have not, a property which I shall call _reflexiveness_; secondly, finite numbers have, while infinite numbers have not, a property which I shall call _inductiveness_. Let us consider these two properties successively.
(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.