User blog:Nayuta Ito/An attempt to well-define OCF as a number notation

'''No set theory. No formal logic. Just be finite.'''

Up to Ω
A string is an erdinal is it's either of the following:
 * 0
 * ψ(A), where A is an erdinal
 * A+B, where A and B are erdinals not being 0

And let the set of all erdinals En.

The standard form of the expression is as follows:

\( f_A(n) \)

,where n is a positive integer and A is an erdinal.

The expression is evaluated as follows. The evaluation always goes from right to left.


 * If the erdinal is 0: \( f_0(n) = n+1 \)
 * Else, if the erdinal is ψ(0): \( f_{ψ(0)}(n) = f_0(f_0(\cdots(f_0(n))\cdots))$, with n f_0's
 * Else, if the erdinal is ψ(A): \( f_{ψ(A)}(n) = f_{<ψ(A)>[n]}(n) \)
 * Else, if the erdinal is A+B:
 * If the erdinal is A+ψ(0): \( f_{A+ψ(0)}(n) = f_A(f_A(\cdots(f_A(n))\cdots))$, with n f_A's
 * Else: \( f_{A+B}(n) = f_{A+  [n]}(n) \)