1. 19
scottaaronson.com
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2. 11

I’d never read the linked post, who can name the bigger number. It’s really good too. http://www.scottaaronson.com/writings/bignumbers.html

1. 5

The sequence of Busy Beaver numbers, BB(1), BB(2), and so on, grows faster than any computable sequence.

I had never understood this property before. Yes, this is worth a read. :)

1. 4

Humanity may never know the value of BB(6) for certain, let alone that of BB(7) or any higher number in the sequence.

Indeed, already the top five and six-rule contenders elude us: we can’t explain how they ‘work’ in human terms. If creativity imbues their design, it’s not because humans put it there.

I think he is on to something very very interesting about computing.

We start from a set of “True Facts” and can find many other things deducible from these facts and axioms.

The interesting thing is the number of true or provably true things and the number of false or provable false things are vanishingly small compare to the things we can compute, yet cannot decide on whether they are true or false.

ie. Being able to decide whether something is true or false is undeniably useful, but is in the smallest class of things we can compute.

The deep question really is, in that vast collection of finite things that are undecidable, which are interesting, or useful or simply beautiful?

1. 5

The interesting thing is the number of true or provably true things and the number of false or provable false things are vanishingly small compare to the things we can compute, yet cannot decide on whether they are true or false.

Agree. And yet it’s also amazing how commonly we encounter feasible problems, given how few they are in number out of “all problems”. That may be an artifact of millennia of experience: we rarely consider infeasible problems, and thus are inclined to structure our model of computation in such a way that if there is a feasible solution, we find it.

Of course, reading code will always be formally undecidable (Rice’s Theorem).