User blog:Primussupremus/Extended busy beaver or super sigma.

The idea for this blog came when I was sitting on the train just a few hours ago reading the large number article by Scott Aaronson,in it he mentions the idea of a super Turing machine a machine can solve the super halting problem. After reading this widely cited article I got the idea for a hiearchy of busy beaver numbers each more powerful than the last. Too start off this blog isn't meant to illustrate any radical ideas the main purpose of it is to expand upon already existing ideas,with that out of the way let us begin. Before we can begin to delve into the hiearchy of busy beavers we need to understand some basic ideas relating to busy beavers,if you already know what these are you can skip this bit. Too start off we need to look at the Busy beaver game invented by Hungarian mathematician Tibor Rado in 1962. The idea behind the Busy Beaver game is that you start with an initially blank Turing machine with n states you then wait for the machine to halt. The Busy Beaver game is essentially trying to find out using a given number of states n how many steps will it take for the Turing machine too finish running. The busy beaver function is uncomputable this means that there is no algorithm that can logically describe it. The busy beaver function is notorious for being down right insane due to the vastness of the outputs produced from even smallish numbers like 5. We generally denote it with two B's like BB or the Greek letter sigma Σ for the sake of argument we will use BB to denote the Busy Beaver function. BB(1)=1 this is by far the most trivial example as a 1 state Turing machine must either halt instantly or go on forever. BB(2)=6 this is one of the first truly interesting numbers in my opinion as it is not only a perfect number but also the only number that when flipped upset down produces a bigger number,this number is of course 9. BB(3)=21 which is also a multiple of 3 not that it really matters. BB(4)=107 if you add the digits together you get 8 which is a multiple of 4. The first 4 Busy Beaver numbers are easy enough to understand but BB5 and beyond is a whole other story. BB(5)≥47,176,870 as you can see the numbers have jumped from 3 digits to 8 digits,although BB(5) could be a lot larger than this seeing as this is just an upper bound. BB(6)≥7.412*10^36,534 this number contains 36535 decimal digits or decits as I like to call them,as you can see from this impressive lower bound on BB(6) the outputs are going to get even crazier. BB(7)≥ 10^10^10^10^10^7. BB(8) has not been given a lower bound although it must be huge compared to Busy Beaver Busy 7. BB(9)is again probably bigger than BB's 7 and 8.The next Busy Beaver is BB(10) this has been proven to be greater than or equal to 3↑↑↑3=3↑↑(3↑↑3)=3↑↑(3^27)=3^3^3^3^3...^3 where there are 3^27 3's in the stack. BB(11) through to BB(7909) have not been given lower bounds but we do know that they must be finite. BB(7910) has been proven to not be able to be determined by the standards axioms of arithmetic ZFC or Zermelo-Fraenkel set theory+the axiom of choice. Using my current knowledge of this disgustingly beautiful game I decided to develop what I call Super Beavers these are Busy beavers where the index of the function is a Busy Beaver. We will use 's like so to denote how many times the input will go into the function for that we need to use one of the oldest tricks in the book recursion. BB(2)(2)=BB(BB(2))=BB(6). BB(3)(3)=BB(BB(BB(3))=BB(BB(21). BB(4)(4)=BB(BB(BB(BB(4)))). BB(n)(n) means that you input n into the function n times too produce a number of such magnitude that it alludes even the greatest minds of the last 10000 years.