Hello everyone! This is Avem231, and welcome to the first part of Journey to Infinity, a resurrection of the Hunt for the Largest Number series that never caught on. Unlike in the previous series, where I simply made an ill-defined salad function, this time, I will actually try to create a clear definition for the googolisms I will be making along the way. Without any further ado, let's begin!
The FGH beyond Γ0[]
In Sbiis Saibian's explanation of the Fast-growing hierarchy, he left off with Γ0 as the largest possible ordinal you can plug into f. Because of this, I will be making my own extension of the FGH that picks up from where he left off, and goes even further beyond with the already-insane potential of the hierarchy.
Diagonalization in the FGH, and how to overcome it[]
There are a lot of infinitely nested power towers used in the FGH to define new ordinals. For example, ε0 is defined as , and ζ0 is defined as . Veblen tried to solve this problem by introducing an ordinal Φ. The defining property of Φ is that when combined with a number (denoted as ), it will represent a given post-ω transfinite ordinal. Φ1, for example, is ε0, Φ2 is ζ0, and so on and so forth. However, it still has the same problem as before, as we can still nest an infinite amount of Φ's to each other to produce Γ0. The problem persists even to the highest levels of the currently known FGH, as ordinals such as ϑ still fall victim to the infinite diagonalization problem. Because of this, I am postulating a new ordinal, designed to finally put an end to the over-usage of infinite power-tower nests in the FGH:
(Note that this article is a work-in-progress. I'll get back to this soon...)