User blog comment:DWither/Doubt about busy beaver/@comment-5529393-20180904212819/@comment-35470197-20180907010231

@DWither

> Then what about an Artificial Super Intelligence that learns from the patterns as they get more complex and makes stronger theories to find them as it progresses through the busy beaver numbers?

If AI uses a single fixed algorithm A to answer whether a given programme eventually halts or not, then there is a programme P such that either one of the following holds: It is irrelevant to whether AI learns in a pattern-matching way or not.
 * 1) The AI costs infinite time to answer whether P halts or not.
 * 2) The AI answers in finite time whether P halts or not, but the answer is false.

It is good to know that there is no reasonable choice of a stronger theory as long as you assume the consistency. For example, if a sentence F is independent of a given theory, then you can add to the theory either one of F and the negation of F as an axiom. If you do not assume the consistency, then you can add 0=1 as an axiom, which gives us the strongest theory.