## FANDOM

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I got this random idea earlier in class and it looked fun so I decided to make a function for it.

## Informal Definition

$Rdc(\alpha, x)$ is a function that takes 2 arguments: an ordinal (it could be a successor ordinal) and a positive integer. It's based on the Fast growing hierarchy[1].

Basically, it's how many steps to take to reduce $\alpha$ to $0$ using $x$ as a base. If you didn't get that, take this for example:

$Rdc(\varepsilon_{0}, 2)$

• $\varepsilon_{0}$ (We don't count this one)
• $\omega^\omega$
• $\omega^2$
• $\omega\times2$
• $\omega + 2$
• $\omega + 1$
• $\omega$
• $2$
• $1$
• $0$ (Victory!)

Therefore, $Rdc(\varepsilon_{0}, 2) = 9$

Let's try $Rdc(\varepsilon_{1}, 2)$

$Rdc(\varepsilon_{1}, 2)$

• $\varepsilon_{0}^{\varepsilon_{0}}$
• $\varepsilon_{0}^{\omega^\omega}$
• $\varepsilon_{0}^{\omega^2}$
• $\varepsilon_{0}^{\omega2}$
• $\varepsilon_{0}^{\omega+2}$
• $\varepsilon_{0}^{\omega+1}\omega^\omega$
• $\varepsilon_{0}^{\omega+1}\omega^2$
• ...

That goes for a long time. I'll post the whole reduction part by part sometime later.

I guess the title was wrong or maybe I just need to do the reduction thing thousands of times to get to 0.

The question is, how "large" are the numbers created by this?