tcai

Yes - they are independent groups with separate projects in different locations (Brain is in Mountain View, DeepMind in London).

I have some issue with how you're generating the BB numbers, though it's not necessarily mathematical. Fundamentally, "cranking it out" is exactly what you cannot do to generate the BBs: following any one specific set of instructions for generating the BBs is a futile exercise.

The problem arises when you say that "a programmer should be able to grind out" whether or not a TM halts or not, which you use to get around the fact that a TM cannot solve the halting problem. However, I'd question if this is a trivial exercise: while we certainly are capable of recognizing infinite loops in the code we write, I'm not certain that humans can identify arbitrary infinite loops. Obviously, whether or not we can isn't a trivially answerable question, as it comes down to whether or not our brain's neural networks can be modeled by a sufficiently large TM, and even if it cannot be modeled by a sufficiently large TM, what differences between our brains and a TM exist and why those would effectively allow us to solve the halting problem.

So I'd question whether finding the BBs is "'just' a matter of computation", because I'm not convinced that humans can solve the TM halting problem.

As peterjmag indicated in his comment, 9^9^9 looks like 9^(9^9) which is actually greater than 9!!. A different way of looking at it is n! < n^n. We can see from here that factorization isn't really the new paradigm that the author is looking for; it's just a part of the exponentiation paradigm.

Furthermore, factorials don't really scale or stack easily. What the author is getting at in the relevant location is stacking the same concept: 1. Multiplying is just adding the same number several times. 2. Exponentiation is just multiplying the same number several times. 3. Tetration is just exponentiation several times. 4. Etc.

This allows us to generate the infinite hierarchy easily expressible by the ackerman numbers (which is basically A(i) = f_i(i,i)), which doesn't generate itself as easily with factorialization in place of exponentiation.