This project is made to compute a very large number which is level ε₁ on the Fast Growing Hierarchy using an extension of Beklemishev's worm. The rules of Beklemishev's worm are as follows: A worm (W) is defined as a string of numbers, e.g (4,3,6). For any worm W, W[n] is defined in the following way: · If W is empty, W[n]=W=(), which should never happen, e.g. ()[6]=() · If the last element of W is 0, remove the last element, e.g. (4,3,6,0)[3]=(4,3,6) · (not in the original definition) If the last element is a limit ordinal, W[n] is W with the last element, λ, replaced with λ[n], e.g. (3,9,ω)[4]=(3,9,4) · If none of the above: -- Let k be the index of the first element that is smaller than the head, if that doesn't exist, k=0. -- Define the good part, g, of the worm to be the every part of the worm up to, including, the k-th element. Define the bad part to be everything after that. -- Decrease the last element of the bad part by one. -- W[n] is concatenation of the good part, followed by n times the bad part, e.g. (1:4,3,3)[2]=(1,4,3,2,4,3,2), ‘:’ is used to visualize the separation between the good and bad part. worm(n) is how long it takes (the value of m at the end) to reduce the worm W=(ωn+n) (originally W=(n)) in the following way: 1. Put m=1 2. Increase m by 1 3. Set W=W[m] 4. If W isn't empty, go back to step 1. The current alphabet of this project is ω², using pairs (a,b)=ωa+b. I might extend it to include normal (only consisting of natural numbers) worms as alphabet, which can bring the growth-rate up to ε(ε(0)). Though I also have another idea which can get me up to φ(ω,0). Click ‘See inside’ to see the project.
