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?