The transition from ‘difficult, but provable’ to ‘mathematics says no’ in busy beaver numbers is one of those great little ah ha! moments. It’s got a similar feeling to the outline proof of Gödel’s incompleteness - that “oh sure, lets just embed a logic in our system of arithmetic, what could possibly go wrong?!” moment.

The transition from ‘difficult, but provable’ to ‘mathematics says no’ in busy beaver numbers is one of those great little ah ha! moments. It’s got a similar feeling to the outline proof of Gödel’s incompleteness - that “oh sure, lets just embed a logic in our system of arithmetic, what could possibly go wrong?!” moment.