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

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It is in a totally different direction that Mr. Zermelo is seeking the solution of the difficulties we have pointed out. He tries to establish a system of axioms a priori, which must allow him to establish all the mathematical truths without being exposed to the contradiction. There are several ways of conceiving the role of axioms; they can be regarded as arbitrary decrees which are only the disguised definitions of fundamental notions. Thus, at the beginning of geometry, Mr. Hilbert introduces “things” which he calls points, straight lines and planes, and that, forgetting or seeming to forget for a moment the vulgar meaning of these words, he puts between these things various relationships that define them.

For this to be legitimate, it must be shown that the axioms thus introduced are not contradictory, and Mr. Hilbert was perfectly successful with regard to geometry, because he supposed the analysis already constituted and that he was able to use it for this demonstration. Mr. Zermelo did not demonstrate that his axioms were free from contradiction, and he could not do so, for to that end he would have had to rely on other truths already established; or already established truths, a science already made, he supposes that there is not yet one, he makes a clean sweep, and he wants his axioms to be entirely sufficient for themselves.

The postulates can not derive their value from a kind of arbitrary decree; they must be self-evident. It will be necessary, therefore, not to demonstrate this evidence, since the evidence does not prove itself, but to seek to penetrate the psychological mechanism which has created this feeling of evidence. And here is where the difficulty comes from; Mr. Zermelo admits certain axioms, and rejects others which, at first sight, may seem as obvious as those which he preserves; if he kept them all, he would fall into the contradiction, he had to make a choice, but we can ask what are the reasons for his choice, and that is what compels us to pay some attention.

Thus he begins by rejecting the definition of Cantor: an ensemble is the union of any discrete objects considered as forming a whole. I do not have the right to speak about all the objects that satisfy this or that condition. These objects do not form an ensemble, a Menge, but we must put something in the place of the definition we reject. Mr. Zermelo limits himself to saying: consider a domain (Bereich) of arbitrary objects; it can happen that between two of these objects x and y, there is a relation of the form x ∈ y; we will then say that x is an element of y, and y is an ensemble, a Menge.

Obviously this is not a definition, someone who does not know what a Menge is, will not know it when he learns that it is represented by the symbol , since he does not know what is. It could go if this symbol was to be defined in the following by the axioms themselves which would be regarded as arbitrary decrees. But we have just seen that this point of view was untenable. We must therefore know in advance what a Menge is, that we have the intuition, and it is this intuition that will make us understand what it is that which would be nothing more than a symbol devoid of meaning, and of which no obvious property could be affirmed by itself. But what can this intuition be if it is not Cantor’s definition that we have disdainfully rejected?

Let us pass on this difficulty, which we shall seek further to clarify and enumerate the axioms admitted by M. Zermelo; they are seven in number:

1° Two Mengen who have the same elements are identical.

2° There is a Menge which contains no element, it is the Nullmenge; if there is an object a, there is a Menge (a) whose object is the only element; if there are two objects a and b, there is a Menge (a, b) of which these two objects are the only elements.

3° The set of all the elements of a Menge M that satisfy a condition x  forms a subset, an Untermenge of M.

4° At each Menge T corresponds another Menge U T, formed of all the Untermengen of T.

5° Consider a Menge T whose elements are themselves Mengen; there exists a Menge S T, whose elements are the elements of the elements of T. If for example T has three elements A, B, C, which are themselves Mengen; if A has two elements a and a’, B two elements b and b’, C two elements c and c’, ST will have six elements a, b, c, a’, b’, c’.

6° If we have a Menge T whose elements are themselves Mengen, we can choose in each of these elementary Mengen an element, and the set of elements thus chosen forms a Untermenge of ST.

7° There is at least one infinite Menge.

Before discussing these axioms, I must answer a question; why, in their statement, did I keep the German name Menge instead of translating it by the word ensemble? It is because I am not sure that the word Menge retains its intuitive meaning in these axioms, without which it would be difficult to reject the definition of Cantor; but the word ensemble together suggests this intuitive sense in a way too imperious, so that it can be used without inconvenience when this meaning is altered.

I will not dwell much on the 7th axiom; I must, however, say a word in pointing out the very original manner in which Mr. Zermelo states it; he does not content himself with the statement I have given; he says: there is a Menge M which can not contain the element a, without also containing as element the Menge (a), that is to say say that whose a is the only element. And so if M admits the element a, it will admit a series of others, namely the Menge of which a is the only element, the Menge whose the only element is the Menge whose unique element is a, and so on. It is clear enough that the number of these elements must be infinite. At first sight, this detour seems very strange and very artificial, and it is indeed; but Mr. Zermelo wished to avoid pronouncing the word infinite, because he considers his axioms as anterior to the distinction between the finite and the infinite.

Let’s move on to the first six axioms; they can be regarded as obvious, as soon as we give the word Menge its intuitive meaning and if we consider only objects in finite number. But they are not more so than this other axiom that the author expressly rejects:

Any objects form a Menge.

And then we have to ask ourselves a question; why does the evidence of axiom 8 cease when it comes to infinite collections, while that of the first six subsists?

If, to solve this question, we refer to the statement of the axioms, we will have a first astonishment; we will find that all these axioms without exception teach us only one thing: certain collections, formed according to certain laws, constitute Mengen; so that these axioms will appear to us only as rules intended to extend the meaning of the word Menge, as pure definitions of words. And this is true of the 8th axiom we reject, as well as the first seven that we accept.

We are warned, however, very quickly that this first impression is misleading; similar definitions of words would not expose us to contradiction; it would only be feared if we had other axioms claiming that some collections are not Mengen; and we do not have any. However, if we reject the 8th axiom, it is to avoid the contradiction: Mr. Zermelo says it explicitly.

It is therefore necessary that he did not regard his axioms as mere definitions of words, and that he attributed to the word Menge an intuitive meaning pre-existing to all his utterances, though somewhat different from the usual meaning. It is not impossible to perceive it by looking for the use that the author makes of it in his reasoning. A Menge is something we can reason about; it is something fixed and immutable to a certain extent. To define an ensemble, a Menge, a collection of any kind is always to make a classification, to separate the objects that belong to this ensemble from those that are not part of it. We will say then that this ensemble is not a Menge, if the corresponding classification is not predicative, and that it is a Menge, if this classification is predicative or if it can be reasoned as if it were.

If we reject the 8th axiom, it is because any objects will undoubtedly form a collection, but a collection which will never be closed, and whose order can be disturbed at any moment by the addition of unexpected elements. It is a collection that is not predicative and on the contrary, when we say for example that at each Menge T corresponds another Menge UT or ST defined in one way or another, we affirm that this definition is predicative, or that we have the right to act as if it were.

And this is the place to speak of a distinction that plays a key role in Zermelo’s theory: “Eine Frage oder Aussage E, ueber deren Gültigkeit oder Ungültigkeit die Grundbeziehungen des Bereiches vermöge der Axiome und der allgemeingültigen logischen Gesetze ohne Willkür unterscheiden, heisst definit.” The word definit here seems to be substantially synonymous with predicative. But the use made by Zermelo shows that synonymy is not perfect. For example, let’s suppose that this question E is the following: does any element of the Menge M have any relation with all the other elements of the same Menge, and which we agree to to say that all the elements for which one must answer yes form a class K? For me, and I also believe for Mr. Russell, such a question is not predicative; because the other elements of M are in infinite number, that we can constantly introduce new ones, and that among the new elements introduced, there may be in the definition of which between the notion of the class K, that is, the ensemble of the elements that have the property E. For Mr. Zermelo, this question would be definit without my knowing exactly where the exact demarcation is, between the questions that are definit and those that are not. It seems to him that, to know if an element has the property E with respect to all the other elements of M, it is enough to check if it possesses it with respect to each one of them. If the question is definit in relation to each of its elements, it will be ipso facto in relation to all these elements.

And it is here that the divergence of our views appears. Mr. Zermelo refrains from considering all the objects which satisfy a certain condition because it seems to him that this ensemble is never closed; that we can always bring in new objects. On the contrary, he has no scruples about talking about all the objects that are part of a certain Menge M and that satisfy a certain condition. It seems to him that he cannot possess a Menge, without at the same time having all his elements. Among these elements, he will choose those who satisfy a given condition, and he will be able to make this choice very quietly, without fear of being disturbed by introducing new and unexpected elements, since these elements, he already has them all between his hands. By posing in advance his Menge M, he raised a wall of fence which stops the intruders who could come from outside. But he does not wonder if there can be inner troubles that he has locked up with him in his wall. If the Menge M has an infinite number of elements, it means not that these elements can be conceived as existing in advance all at once, but that it can constantly create new ones; they will be born inside the wall, instead of being born outside, that’s all. When I speak of all integers, I mean all the whole numbers that we have invented and all those that we can invent one day; when I speak of all the points of space, I mean all the points whose coordinates are expressible by rational numbers, or by algebraic numbers, or by integrals, or in any other way that we can invent. And it is this “one will be able” which is the infinite. But we can invent that we will define in many ways, and if we resume as our earlier question E and our class K, the question E arises again each time we define a new element of M; now, among these elements that we will be able to define, there will be some whose definition will depend on this class K. So that the vicious circle could not be avoided.

That is why Mr. Zermelo’s axioms would not satisfy me. Not only do they not seem obvious to me, but when I am asked if they are free of contradiction, I will not know what to say. The author thought to avoid the paradox of the greatest cardinal, by refraining from any speculation outside the enclosure of a well-closed Menge; he thought he avoided Richard’s paradox by asking only definite questions, which, according to the meaning he gives to this expression, excludes any consideration of objects that can be defined in a finite number of words. But if he has closed his sheepfold, I am not sure that he has not locked up the wolf. I would be calm only if he had shown that he was safe from contradiction; I know very well that he could not do it, since it would have been necessary to rely for example on the principle of induction, which he did not revoke in doubt, but which he proposed to demonstrate further. He should have gone over it; it would have been at the price of a fault of logic, but at least we would be sure of it.

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