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).

翻译：我们针对稳定可计算可枚举形式系统，证明了与哥德尔第一和第二不完备性定理直接对应的类比结果。典型的稳定可计算可枚举集合是无整数解的丢番图方程集合，特别地，这类集合通常不是可计算可枚举的。因此，这为第二不完备性定理提供了首次向非经典可计算形式系统的扩展。让我们通过一个具有物理背景的应用来阐明其意义。令 $\mathcal{H}$ 表示人类数学产出在无限时间极限下的适当结果（即本文意义上的稳定化），特指算术语言中的一阶语句（为简化起见），并将其理解为一个形式系统。假设 $\mathcal{H}$ 形成过程中所有相关物理过程都是图灵可计算的。那么根据定义，$\mathcal{H}$ 可能*不*是可计算可枚举的，但它是稳定可计算可枚举的。因此，应用于 $\mathcal{H}$ 的经典哥德尔析取命题将失去意义，但将我们的不完备性定理应用于 $\mathcal{H}$ 时，我们得到了哥德尔析取命题的一个强化版本：假设 $\mathcal{H} \vdash PA$，则要么 $\mathcal{H}$ 不是稳定可计算可枚举的，要么 $\mathcal{H}$ 不具备1-一致性（特别地不具备可靠性），要么 $\mathcal{H}$ 无法证明某个为真的算术陈述（若 $\mathcal{H}$ 额外满足2-一致性，则也无法证伪该陈述）。