Ackermann was born on March 29, 1896 in Schönebecke (then Altena district, now part of Herscheid municipality), Germany, and received his doctoral degree in 1925 with his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic without induction. From 1929 until 1948 he taught at the Arnoldinum high-school in Burgsteinfurt, and afterwards till 1961 in Lüdenscheid. He was also a corresponding member of the Akademie der Wissenschaften (Academy of Sciences) in Göttingen, and was an honorary professor at the Universität Münster (Westphalia).
He wrote Grundzüge der Theoretischen Logik (Fundamentals of the Theoretical Logic) with David Hilbert, dealing with Entscheidungsproblem and also constructed consistency proofs for set theory (1937), full arithmetic (1940), type-free logic (1952) and a new axiomatization of set theory (1956). He wrote the book Solvable cases of the decision problem (North Holland, 1954).