# Was Hilbert ambivalent about set theory?

There is the well-known quote of Hilbert: "No one shall drive us from the paradise which Cantor has created for us." [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167] On the other hand Hilbert concludes his paper:

Finally we will return to our original topic and draw the conclusions of all our investigations about the infinite. On balance the result is this: The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking – a remarkable harmony of being and thinking. [loc cit, p. 190]

Further Hilbert devised "Hilbert's hotel": An infinite hotel is completely filled with guests. Another guest arrives. He gets room no. 1 after every resident guest has moved on by one number from $$n$$ to $$n+1$$. Even infinitely many guests can be accommmodated when every resident guest doubles his room number.

Did Hilbert set up this example in order to counter Cantor's list?

Hilberts infite hotel is really infinite, unfinished, extendable but Cantor's list is not. Two different interpretations of one and the same infinity.

Only that allows to conclude that the antidiagonal, as a new guest differing from all resident guests or entries of the list, cannot be inserted, for instance into the first position when every other entry moves on by one "room number". Without Cantor's arbitrary constraint even all infinitely many antidiagonals that ever could be constructed could be accommodated. Cantor's theorem would go up in smoke.

My question: Is there evidence that Hilbert intended this counter argument? Or did he not realize that it is a counter argument? But then what was his purpose?

• Frankly speaking, I do not think taht a first-class mathematician like Hilbert ever hold such simple contradictory statement. I would prefer to assert that we do not read correctly his words. Feb 19, 2018 at 13:46
• I'm not sure I understand the point being made here. In the diagonal argument, "non-extendability" is our hypothesis, not an "arbitrary constraint". Cantor is using his definition of countable. If $\mathbb R$ is countable, then there exists a "list" of all real numbers. By hypothesis, this list cannnot be extended because it is a "complete" list containing all of the real numbers so there are no antidiagonals to be added.
– nwr
Feb 19, 2018 at 19:12
• @ Mauro ALLEGRANZA: You can be sure that Hilbert said precisely these words. Feb 19, 2018 at 20:56
• @Nick R: Non-extendability is our hypothesis? Only concerning the enumeration of the list. The real numbers are extended by producing the antidiagonal number. If this real number had been existing already when creating the list, we would certainly have inserted it, wouldn't we? If not that amounts to deliberately cheating. And of course all antidiagonals ever produced belong to a countable set and could be accomodated in a single list like all new guests in Hilbert's hotel. Feb 19, 2018 at 21:01

## 1 Answer

No, he was not, as one can see from the full passage from Hilbert's lecture On the Infinite delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, and published in Mathematische Annalen vol. 95 (1926):

"In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought — a remarkable harmony between being and thought. In contrast to the earlier efforts of Frege and Dedekind, we are convinced that certain intuitive concepts and insights are necessary conditions of scientific knowledge, and logic alone is not sufficient. Operating with the infinite can be made certain only by the finitary.

The role that remains for the infinite to play is solely that of an idea — if one means by an idea, in Kant's terminology, a concept of reason which transcends all experience and which completes the concrete as a totality — that of an idea which we may unhesitatingly trust within the framework erected by our theory."

"An idea which we may unhesitatingly trust" does not exactly sound ambivalent. Hilbert was a formalist, not a platonist, he did not believe that "actually infinite" actually exists, or that it needs to exist to be talked about. Being a Kantian regulative idea ("noumenon"), finitely axiomatized into a formal theory, was more than enough for mathematics, according to him. The only requirement is that the said theory be consistent, and the whole lecture was to promote the so-called Hilbert programme of proving consistency of infinitary theories by finitary means.

It should be clear from this that Hilbert's hotel is neither real, nor meant to counter Cantor. What Cantor did or did not assume in the diagonal argument is moot since it is derivable in finitely many steps from set-theoretic axioms. Inspection of Hilbert's earlier 1924 lecture, where it is described (along with a similarly minded infinite dance party and a quip "in a world with an infinite number of houses and occupants there will be no homeless"), confirms it. It was a smalltalk introduction of the difference between the finite and the infinite aimed at lay audience, it is not connected to Cantor, discussed or even mentioned in that lecture, or any other, after being introduced. Here is from The True (?) Story of Hilbert’s Infinite Hotel by Kragh:

"It was merely an example and one that he attached no particular importance to. Nor did other people at the time find it important. Had the hotel not been resuscitated by Gamow more than two decades later it might well be unknown today. The only allusion to it before 1947 that I know of is from a textbook on calculus published in 1938 and written by Otto Haupt, a mathematician at Erlangen University."