As I understand it, the game around functions like this is to get as close to infinity as you can, but not quite, and then to try to uncover properties about what you find there.
I'm under the impression that it's a certain kind of fun because the results are all way too large to work with computationally, so rather than comparing the values (which you can't calculate directly) you have to reason about the various algorithms that yield them.
That's all I got. The gust of the post is greek to me. I wish I had more computer science and less software engineering under my belt. Then maybe this could be my kind of fun too.
Well, the goal is to get as close to infinity as possible with the smallest program possible. We can name big numbers just fine, by recursing over recursion for some number of steps, but the fun part is to have these fall out 'naturally' from the space of small Turing machines.
In this case, we can actually name the number of symbols in terms of Knuth's up-arrow notation [0], which describes the sequence of operations starting with multiplication, exponentiation, repeated exponentiation (called tetration, e.g., 2↑↑5 = 2^[2^[2^[2^2]]]), repeated tetration (called pentation, e.g., 2↑↑↑5 = 2↑↑[2↑↑[2↑↑[2↑↑2]]]), and so on. This number is big enough that it needs 15 arrows in a row to describe it reasonably. So it's not just that the number is very large, it's that we can also put a neat lid over its value. For instance, we know it's still nothing compared to Graham's number, which can't be described with any reasonable number of up-arrows.
https://en.wikipedia.org/wiki/Busy_beaver
https://en.wikipedia.org/wiki/Ackermann_function
As I understand it, the game around functions like this is to get as close to infinity as you can, but not quite, and then to try to uncover properties about what you find there.
I'm under the impression that it's a certain kind of fun because the results are all way too large to work with computationally, so rather than comparing the values (which you can't calculate directly) you have to reason about the various algorithms that yield them.
That's all I got. The gust of the post is greek to me. I wish I had more computer science and less software engineering under my belt. Then maybe this could be my kind of fun too.