# 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. Commented 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
Commented Feb 19, 2018 at 19:12
• @ Mauro ALLEGRANZA: You can be sure that Hilbert said precisely these words. Commented 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. Commented Feb 19, 2018 at 21:01