User blog:King2218/The Slow Rdc(a, x) Function

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.

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. What is cool about the reduction above is that every 4 steps, a term has a factor of $$\omega2$$ which is pretty cool.

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?