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Henri Poincaré, The logic of infinity: The cardinal number

We must not forget the preceding considerations when defining the cardinal number. If we consider two collections, we may seek to establish a law of correspondence between the objects of these two collections, so that any object of the first corresponds to an object of the second and only one, and vice versa. If possible, both collections are said to have the same cardinal number.

But here again, this law of correspondence must be predicative. If we are dealing with two infinite collections, we can never conceive of these two collections as exhausted. If we suppose that we have taken in the first a certain number of objects, the law of correspondence will enable us to define the corresponding objects of the second. If we then introduce new objects, it may happen that this introduction changes the meaning of the law of correspondence, so that the object A of the 2nd collection, which before this introduction corresponded to an object A of the 1st, will not correspond any more after this introduction. In this case the law of correspondence will not be predicative.

And that is what we will explain by two opposite examples. I consider the set of integers and the set of even numbers. At each integer n I can match the even number 2n. When I introduce new integers, it will always be the same number 2n which will correspond to n. The law of correspondence is predicative, and the same is true of all those that envisage Cantor to demonstrate for example that the cardinal number of rational numbers is equal to that of integers, or that of the points of space to that of points of a line.

Suppose instead that we compare the set of integers to that of the points of space that can be defined by a finite number of words and that I establish the following correspondence between them. I will make the table of all the possible sentences, I will order them according to the number of their words, by arranging in alphabetical order those which have the same number of words. I will erase all those that have no meaning or define no point, or that define a point already defined by one of the preceding sentences. I will correspond to each point the sentence that defines it, and the number that occupies this sentence in the table thus pruned.

When I introduce new points, it may happen that phrases which are meaningless acquire one; they must be restored to the picture from which they were first erased; and the number of all the other sentences will be modified. Our correspondence will be completely upset; our law of correspondence is not predicative.

If one did not pay attention to this condition in the comparison of the cardinal numbers, one would be led to singular paradoxes. It is therefore necessary to modify the definition of cardinal numbers by specifying that the law of correspondence on which this definition is based must be predicative.

Any law of correspondence is based on a double classification. We must classify the objects of the two collections that we want to compare; and both classifications must be parallel; if, for example, the objects of the 1st are divided into classes, which are subdivided into orders, these into families, etc., the same must be the case with the second. To each class of the 1st classification will have to correspond a class of the 2nd and only one, to each order an order and so on, until one reaches the individuals themselves.

And then we see what must be the condition for a law of correspondence to be predicative. The two classifications on which this law is based must themselves be predictive.

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