Not long ago I finded Goodstein Functions pretty amazing, and it lead me to make this function. The Function is expressed as B(j)=k, where j is the seed, which is the number we're dealing with, and k is the length of the sequence – the # of steps it takes to reach 0. It is hard explain, but I shall my best.

I use square brackets around the number in any base b, and space between digit to avoid confusion when the base is higher than 10. It means for instance that [1 2]_{13} ≠ [12]_{13}, since the first one is equal to 15 and the second one is equal 12, and therefor not the same.

The seed number is what we are starting, and it need to be written in base 2. Example: if the seed is 3 then you write it as 11 in base 2 or in the Batrix way of writing it: [1 1]_{2}. The main thing is to find out long this seed number take to reach 0 through a process. The process is as following: increase the base by 1, then increase every number that was one less than the previous base by 1, afterward -1 the number (not the base).

j=1

Base | Batrix notation | value |
---|---|---|

2 | [1]_{2}
| 1 |

3 | [1]_{3}
| 1 |

4 | [0]_{4}
| 0 (stop here) |

Length is 3, therefor we can conclude B(1)=3

Let us try j=2 and see what it get us to.

Base | Batrix notation | Value |
---|---|---|

2 | [1 0]_{2}
| 2 |

3 | [1 2]_{3}
| 5 |

6 | [1 0]_{6}
| 6 |

8 | [6]_{8}
| 6 |

14 | [0]_{14}
| 0 |

B(2)=13

Let's do j=5

Base | Batrix notation | Value |
---|---|---|

2 | [1 0 1]_{2}
| 5 |

3 | [2 0 1]_{3}
| 19 |

4 | [3 0 0]_{4}
| 48 |

5 | [3 4 4]_{5}
| 99 |

10 | [3 9 0]_{10}
| 390 |

12 | [3 8 10]_{12}
| 538 |

12*2^{9}
| [2 12*2^{9}-1 12*2^{9}-2]_{12*29}
| |

12*2^{12*29+9}=a
| [1 a-1 a-2]_{a}
| |

12*2^{a+12*29+9}=b
| [b-1 b-2]_{b}
| |

12*2^{b+a+12*29+9}-2=c
| [0]_{c}
| 0 |

B(5)=c-1, and c is gigantic number which we can compare in FGH as being less than or equal to f_{2}^{5}(12).

Already at 5 it's pretty huge. These I showed you are just a little example on a level 1 Batrix Function. The level 2 also include more rows, just like in matrix. It can reach f_{ω2}-scale.

Source: https://djgrexnumbers.wordpress.com/2016/11/23/batrix-function/