Two pages from the article in Mathematische Annalen 95:
David Hilbert "Über das Unendliche",
1925.
Hilbert's 1925 lecture (online) stops short
of the part on double recursive functions.
For a full English translation of the article,
with an introduction, see:
Jean van Heijenoort "From Frege to Gödel", 1967,
pp 367-392 "On the infinite".