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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?

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