How does Kurt Gödel's work on Incompleteness relate to Ackermann's function? If at all

Question

I know their work was published relatively around the same time ( Gödel's in 1931 and Ackermann's in 1928). Were they related in any way ?

I also know Gödel properly defined Primitive Recursive functions in his proofs and Ackermann's function is an example of a function that's total and computable but not primitive recursive. Doesn't that mean he had to use some sort of definition for a primitive recursive function ? I'm just a little confused because of the time difference and how closely related they are.

Answer 1

The link between Gödel and Ackermann's work is quite strong.

Gödel's work was dedicated to the basic open problems of Hilbert's school : the completeness of predicate calculus and the consitency of arithmetic and analysis.

The problem of the completeness of first-order logic was posed by Hilbert and Ackermann (1928) and the result was obtained by Gödel in 1929.

After proving the completeness of first-order logic, Gödel set to work on proving the consistency of analysis. Instead of directly giving a finitistic proof of analysis, Gödel attempted to first reduce the consistency of analysis to that of arithmetic.

Wilhelm Ackermann was a prominent collaborator of Hilbert.

Ackermann 1925 dissertation : Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit:

is a milestone in the development of proof theory. The work contains the first unified presentation of a system of second-order arithmetic based on the ε-calculus, a complete and correct consistency proof of the ε-less fragment (an extension of what is now known as primitive recursive arithmetic, $\mathsf {PRA}$), and an attempt to extend Hilbert’s ε-substitution method to the full system.

Thus, Ackermann thesis contained the full "machinery" of the primitive recursive class of functions and as well as the method that allowed him in 1928 to go beyond it.

See :

• Paolo Mancosu, The Adventure of Reason : Interplay Between Philosophy of Mathematics and Mathematical Logic. 1900-1940 (2010).

Answer 2

Primitive recursion was actually originally introduced by Dedekind in 1888, and proposed as a basis for arithmetic by Skolem in 1923. Gödel didn't define primitive recursion - he devised a way of encoding it inside arithmetic, so that Peano Arithmetic could "do" recursion, which was an important step in the Incompleteness theorem.

To my knowledge, there's no connection between Gödel's work and Ackermann's function closer than that.