Born on March 29, 1896 in Schoenebeck (Kr. Altena), Germany, recieved his doctoral degree in 1925 with his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreihei, which was a consistency proof of arithmetic without induction.

From 1927 until 1961 he taught at the high-schools in Burgsteinfeld and in Luedenscheid.

He was a corresponding member of the Akademie der Wissenschaften (Academy of Sciences) in Göttingen, and was an honorary professor at the Universität Münster (University of Cathedral).

In 1928, he 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. It was later simplified by Rosza Peter whose initial condition was simplified by Raphael Robinson and is now known as the Ackermann function

He also wrote Grundzuege der Theoretischen Logik (Fundamentals of the Theoretical Logic) with David Hilbert.

Ackermann also wrote consistency proofs for set theory (1937), full arithmetic (1940), type free logic (1952), a new axiomatization of set theory (1956), and a book Solvable cases of the decision problem (North Holland, 1954).

He died in Luedenscheid, Germany on December 24, 1962.