This is probably basic. I am parsing Godel's letter to von Neumann and I am stuck in the following passage where he says "maxF"
One can obviously easily construct a Turing machine, which for every formula F in first order predicate logic and every natural number n, allows one to decide if there is a proof of F of length n (length = number of symbols). Let Ψ(F, n) be the number of steps the machine requires for this and let ϕ(n) = maxF Ψ(F, n).
I understand that Ψ(F, n) is the function of how many steps are required to decide if there is a proof of F with a length of n. I don't understand what function ϕ(n) represents because I'm not sure what he means by maxF.
Complete letter: https://www.anilada.com/notes/godel-letter.pdf