Old system ($$\omega^{\omega^{\omega}}$$):

New system ($$\varepsilon_{0}$$):

If you don't mind me asking. What is epsilon-0? I just searched it up and then my peanut sized brain doesn't understand it.

An infinite power-tower of Omegas. What I mean by "Epsilon-0" is I made a function that grows as quickly as $$f_{\varepsilon_{0}}(n)$$ in the fast growing hierarchy. For reference, $$f_{\omega+1}(64)$$ is above graham's number, $$f_{\omega^2}(n)$$ is the limit of Conway Chained Arrow notation, $$f_{\omega^\omega}(n)$$ is the limit of BEAF, and $$f_{\omega^{\omega^\omega}}(n)$$ is the limit of the old version.

Here is the definition of the Fast Growing Hierarchy:

$$f_0(n) = n+1$$

$$f_{\alpha+1}(n) = f_{\alpha}^n(n)$$ (function iteration)

$$f_\alpha(n) = f_{\alpha[n]}(n)$$ (depends on fundamental sequences)

Wainer Hierarchy (The one that says how epsilon 0 works)

$$\alpha+1 = \alpha\cup\left\lbrace\alpha\right\rbrace$$

$$\omega[n] = n$$

$$\omega^{\alpha+1}[n] = \underbrace{\omega^\alpha+\omega^\alpha+\omega^\alpha+\cdots}_n (\alpha \in \text{Suc})$$

$$\omega^\alpha[n] = \omega^{\alpha[n]} (\alpha \in \text{Lim})$$

$$\varepsilon_0[0] = 0$$

$$\varepsilon_0[n+1] = \omega^{\varepsilon_0[n]}$$