G\"odel Incompleteness Theorem for PAC Learnable Theory from the view of complexity measurement

Abstract

Different from the view that information is objective reality, this paper adopts the idea that all information needs to be compiled by the interpreter before it can be observed. From the traditional complexity definition, this paper defines the complexity under "the interpreter", which means that heuristically finding the best interpreter is equivalent to using PAC to find the most suitable interpreter. Then we generalize the observation process to the formal system with functors, in which we give concrete proof of the generalized G\"odel incompleteness theorem which indicates that there are some objects that are PAC-learnable, but the best interpreter is not found among the alternative interpreters. A strong enough machine algorithm cannot be interpretable in the face of any object. There are always objects that make a strong enough machine learning algorithm uninterpretable, which puts an upper bound on the generalization ability of strong AI.