Incompleteness for stably consistent formal systems
This work addresses foundational issues in logic and computability, potentially impacting AI and intelligence theory, but it appears incremental as it builds on Gödel's theorems with a new consistency notion.
The authors tackled the problem of formalizing human-like consistency in mathematical systems, resulting in a generalization of Gödel's incompleteness theorems to stably consistent systems, which re-proves the original theorems using Turing machines without relying on the diagonal lemma.
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 Gödel 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 "Gödel 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 Gödel, obstructing computability of intelligence.