  We first partly develop a mathematical notion of stable consistency intended
to reflect the actual consistency property of human beings. Then we give a
generalization of the first and second G\"odel incompleteness theorem to stably
$1,2$-consistent formal systems. Our argument in particular re-proves the
original incompleteness theorems from first principles, using Turing machine
language to (computably) construct our "G\"odel sentence" directly, in
particular we do not use the diagonal lemma, nor any meta-logic, with the proof
naturally formalizable in set theory. In practice such a stably consistent
formal system could be meant to represent the mathematical output of humanity
evolving in time, so that the above gives a formalization of a famous
disjunction of G\"odel, obstructing computability of intelligence.
