Gödel doesn't solve constructivity problem
If Gödel’s theorem really refuse mechanism, it will also show how to solve the constructivity problem in the foundations of mathematics. But Gödel's theorem does not how to solve that problem.
Judson Webb (1968).
The Constructivity Problem
The problem of whether or not the foundations of mathematics can be formulated purely constructively, that is, without recourse to assumptions about a platonic ontology of abstract mathematical entities.
If mathematics could be founded on a purely constructional basis, that would mean that it can be constructed "mechanistically".