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Henri Poincaré, The logic of infinity: The memory of Mr. Russel

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Russell published in the American Journal of Mathematics, vol. XXX, under the title Mathematical Logics as Based on the Theory of Types, a memoir in which he relies on considerations quite similar to those which precede. After recalling some of the most famous paradoxes among logicians, he looks for their origin and he rightly finds it in a sort of vicious circle. We have been led to antinomies because we have considered collections, containing objects in the definition of which there is the notion of the collection itself. Non-predictive definitions were used; we have confounded, says M. Russell, the words all and any, which we can render in French by the words tous and quelconque.

He is thus led to imagine what he calls the hierarchy of types. Be a true proposition of any individual of a given class. By any individual, we must first hear all the individuals of this class that can be defined without using the notion of the proposition itself. I will call them any individuals of the first order; when I affirm that the proposition is true of all these individuals, I will affirm a proposition of the first order. Any individual of the second order will then be an individual in whose definition the notion of this proposition of the first order can be introduced. If I affirm the proposition of all the individuals of the 2nd order, I will have a proposition of the 2nd order. The individuals of the 3rd order will be those in the definition of which the notion of this proposition of the 2nd order can be involved; And so on.

Take the example of Epimenides. A liar of the first order will be the one who always lies except when he says I am a liar of the first order; a second-order liar will be the one who always lies when he says I’m a first-order liar, but who does not lie when he says I’m a second-order liar. And so on. And then when Epimenides tells us: I am a liar, we can ask him: of what order? And it is only after he has answered this legitimate question that his statement will make sense.

Let’s move on to a more scientific example and consider the definition of the whole number. We say that a property is recursive if it belongs to zero, and if it can not belong to n without belonging to n + 1; we say that all numbers that have a recurring property form a recursive class. Then an integer is by definition a number which has all the recurrent properties, that is to say which belongs to all the recurrent classes.

From this definition can we conclude that the sum of two integers is an integer? It seems so; for if n is an integer, given, the numbers x which are such that n + x is integer form a recurring class. The number x would not be integer, if n + x was not integer. But the definition of this recurrent class of which we have just spoken is not predicative, because in this definition (which teaches us that n + x must be integer) between notion of integer which presupposes the notion of all recurrent classes.

Hence the need to use the following detour: let us call recursive classes of the first order all those that can be defined without introducing the notion of integer, and integers of the first order the numbers which belong to all the recurrent classes of the 1st order; let us then call recurring classes of the second order those which can be defined by introducing, if necessary, the notion of integer of the first order, but without involving the notion of integer of higher order; let’s call integers of the 2nd order the numbers that belong to all the recurring classes of the 2nd order, and so on. And then what we can demonstrate is not that the sum of two integers is an integer, it is that the sum of two integers of order K, is an integer of order K – 1.

These examples were enough, I think, to explain what Mr. Russel calls the hierarchy of types. But then there are various questions on which the author has not spoken.

1° In this hierarchy, propositions of the 1st, 2nd order, etc., and in general of the n order are introduced without difficulty, ne being any finite integer. Is it possible to consider also propositions of order α, where α is a transfinite ordinal number? Thus Mr. König has imagined a theory which does not differ essentially from that of Mr. Russell; he uses a special notation, he denotes by A(NV) the objects of the first order, by A(NV)2 those of the 2nd order, etc., NV being the initials of the expression ne varietur. As for him, he does not hesitate to introduce A(NV)α where α is transfined, without explaining enough what he means by that.

2° If we answer yes to the first question, we must explain what we mean by objects of order ω, ω being the ordinary infinity, that is to say the first transfinite ordinal number, or with objects of order α, α being any transfinite ordinal.

3° If, on the contrary, we answer no to the first question, how can the distinction between finite or infinite numbers be founded on the theory of types, since this theory is meaningless if we assume this distinction already made.

4° More generally, whether or not we answer the first question, the theory of types is incomprehensible, if we do not suppose the theory of ordinals already constituted. How can one then base the theory of ordinals on that of types?

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