Paul Lévy on the theorems of Gödel and Cohen

In 1931 Kurt Gödel proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. His masterpiece Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940) is a classic of modern mathematics. In this he proved that if an axiomatic system of set theory of the type proposed by Russell and Whitehead in Principia Mathematica is consistent, then it will remain so when the axiom of choice and the generalised continuum-hypothesis are added to the system. This did not prove that these axioms were independent of the other axioms of set theory, but when this was finally established by Paul Cohen in 1963, Cohen built on these ideas of Gödel.

Paul Lévy's approach to mathematics was highly intuitive so it is not surprising that he found himself at odds with the theorems of Gödel and Cohen. He wrote to Jean Dieudonné giving his ideas about these theorems and asking Dieudonné to give his opinions on them.

The following, published in 1968, is an English translation of Lévy's reply after receiving Dieudonné's opinions. The original was published 'Observations de M Paul Lévy sur la lettre de M Dieudonné', Revue de Métaphysique et de Morale 73 (2) (1968), 250-251.

Observations by M Paul Lévy on the letter from M Dieudonné.

I first want to thank M Dieudonné for having at my request made known his opinion on the theorems of Gödel and Cohen, and on my interpretation of these theorems. I'm not surprised that it does not coincide with mine. It was precisely because I knew I did not agree with the specialists in axioms, and that he was particularly able to explain their point of view to me, that I asked him to do so. I would now like to say what, despite these explanations, prevents me from rallying to this point of view.

M Dieudonné is surprised that I believe in a reality of which the demonstrable is only a part. Of course, I am speaking of a mathematical reality, unrelated to that of the physical world, and I am willing to admit that the mathematical object is a convenient fiction, which contains nothing more than what results from the axioms used to define it. Such is, in particular, the case with the set of real numbers. But this set is uncountably infinite, which means that it contains elements that we can never isolate from others and examine individually, even if we could devote a trillion centuries to this task. Defined no doubt on the basis of axioms, it has become a reality of which we will never be able to know more than a tiny part. So there are theorems that are true, and not provable, and I am inclined to believe that there are some which can still be stated in a finite number of words. The work of Gödel and Cohen can only confirm me in this opinion.

It is moreover easy to clarify my idea on examples which are simpler than the theorem of the continuum. Let us consider for this purpose the following statements: "Euler's constant is a transcendent number; the number π has an infinity of decimal places equal to 5; in the sequence of odd numbers, it happens an infinite number of times that two consecutive numbers are prime." For each of these statements, it will not occur to any mathematician that it can be sometimes true, sometimes false. It is easy to multiply examples of this kind. But who would dare to assert that, for each of them, we can arrive at knowing whether it is true or false? Infinity is not contained in the finite. The infinite series of verifications that would have to be carried out cannot undoubtedly always be replaced by reasoning which can be expressed in a finite number of words. So, again, the true, but unprovable, statement is a logical possibility.
It seems to me, on the other hand, that we must distinguish two kinds of axioms, which axiomatists tend to confuse, by attributing to all the character of arbitrary conventions which one could not make. I refuse to attribute this character to a statement such as "if A is a part of B, which is itself a part of C, then A is a part of C". For me this is a necessary consequence of the definition of the word "part", and not a new axiom. I will say the same thing of the one quoted by M Dieudonné "there is a set and only one containing two given elements and containing no other". Isn't this an obvious consequence of the fact that, by definition, a whole is the union of its elements?

If I understand correctly, the axiom is for axiomatists a rule allowing a certain operation. To me, some of these rules are arbitrary, while the others are imposed by the nature of things (or, if you prefer, by previous definitions). I imagine that the axiom systems of Gödel and Cohen are of the second type, while that of von Neumann-Gödel, the foundation of set theory, is of the first type. It only contains, I think, axioms which we cannot reject. But I ask two questions about it: do the finite combinations of applications of these axioms give everything we can know about sets? What we can know does it exhaust all the true?

The answer to the first question may be in the affirmative. But it would have to be demonstrated in order to be able to affirm that the correctness and the falsity of the continuum hypothesis are equally unprovable, and I do not see how it would be demonstrated. Perhaps tomorrow we will discover a new axiom, incompatible with Cohen's system, and which our successors will take as obvious (such as the axiom of choice, which no one had spoken of before Zermelo, and which is now admitted by almost all mathematicians).

However, I am quite inclined to believe that the continuum hypothesis is really undecidable to human logic (I only doubt that this is really demonstrated). But then the answer to my second question can only be negative. Since this hypothesis can only be true or false, the knowable does not exhaust all the true.
What interested me the most in M Dieudonné's presentation is that it makes Gödel and Cohen's ideas, I will not say clear to me, but a little less obscure. I am not embarrassed by the use of models, and I think all the less of contesting their advisability since it was with the help of a model that Poincaré demonstrated that Lobachevsky's geometry could not lead to any contradiction. But I do not see how the fact that Gödel and Cohen based their reasoning on model building should lead me to change my conclusions. A model proves nothing if it is not a faithful image of the object it wants to represent. However Cohen defined a set of an intermediate power between the countable and the continuous, I can conclude that there is a set of real numbers with such a power, and I cannot avoid the contradiction with the conclusion of Gödel that by admitting that their axioms are rules of calculation which, as I said above, only allow to reach a part of the true. So I fall back on the conclusion of my previous article: Gödel and Cohen have solved, one in the affirmative, the other in the negative, problems which are not identical to Cantor's.

On the other hand, I agree with M Dieudonné in thinking that we should not exaggerate the importance of our discussion. Mathematics, and in particular axiomatics, will develop without taking this into account. But I think that, for philosophers, it is interesting to know if mathematical reality exists, in the sense that I indicated above. If we admit it, we can hardly avoid concluding that the knowable is only a part of the true, and admitting my conclusion about the hypothesis of the continuum.

Paul Lévy.

Last Updated September 2020