David Hilbert - double recursion by φn(a,b)

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".

D. Hilbert - Über das Unendliche - Seite 185 D. Hilbert - Über das Unendliche - Seite 186