  We prove, for stably computably enumerable formal systems, direct analogues
of the first and second incompleteness theorems of G\"odel. A typical stably
computably enumerable set is the set of Diophantine equations with no integer
solutions, and in particular such sets are generally not computably enumerable.
And so this gives the first extension of the second incompleteness theorem to
non classically computable formal systems. Let's motivate this with a somewhat
physical application. Let $\mathcal{H} $ be the suitable infinite time limit
(stabilization in the sense of the paper) of the mathematical output of
humanity, specializing to first order sentences in the language of arithmetic
(for simplicity), and understood as a formal system. Suppose that all the
relevant physical processes in the formation of $\mathcal{H} $ are Turing
computable. Then as defined $\mathcal{H} $ may \emph{not} be computably
enumerable, but it is stably computably enumerable. Thus, the classical G\"odel
disjunction applied to $\mathcal{H} $ is meaningless, but applying our
incompleteness theorems to $\mathcal{H} $ we then get a sharper version of
G\"odel's disjunction: assume $\mathcal{H} \vdash PA$ then either $\mathcal{H}
$ is not stably computably enumerable or $\mathcal{H} $ is not 1-consistent (in
particular is not sound) or $\mathcal{H} $ cannot prove a certain true
statement of arithmetic (and cannot disprove it if in addition $\mathcal{H} $
is 2-consistent).
