Wilhelm Ackermann received his doctoral degree in 1925 with a thesis written under Hilbert. Its content was a consistency proof of arithmetic without induction. From 1927 until 1961 he taught as a teacher at the Gymnasien in Burgsteinfeld and in Luedenscheid. He was corresponding member of the Akademie der Wissenschaften in Göttingen, and was honorary professor at the UniversitŠt Münster. He worked on logic with Hilbert.

In 1928, Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive. A(x,y,z) was simplified to a function P(x,y) of 2 variables by Rosza Peter whose initial condition was further simplified by Raphael Robinson. This last function is called Ackermann's function in today's textbooks. Also in 1928, Hilbert and Ackermann coauthored the logic book *Grundzuege der Theoretischen
Logik*.

Among Ackermann's later work there are consistency proofs for set theory, full arithmetic , and type free logic. He also gave a new axiomatization of set theory in 1956, and wrote the book *Solvable cases of the decision problem* in 1954.