User blog:Nayuta Ito/An April Fools Googologism

Spoiler: This number is well-defined.

Bigger spoiler: Everything on this page is a truth.

Even bigger spoiler: This is just a joke.

Definitions
X and X' are strings. Y is 0 or more "]"s. n and A are natural numbers (except zero). If no rule can be applied, that means the number is undefined. X' should have matched brackets.

o n = n X-o n = X (n+1) X[]Y n = XY n X[X'-o]Y n = X[X'][X']…[X']Y n /w n+1 "[X']"s X[X'<0>o]Y n = X[X'-o[-o[…[-o]]]]Y n /w n "-o"s X[X'o]Y n = X[X'o[o[…[o]]]]Y n /w n s

Examples
o-o[-o-o[<1>o]]-o-o 2 =o-o[-o-o[<1>o]]-o 3 =o-o[-o-o[<1>o]]4 =o-o[-o-o[<0>o[<0>o[<0>o[<0>o]]]]]4 o-o-o-o-o[<4>o] 3 =o-o-o-o-o[<3>o[<3>o[<3>o]]] 3 o-o[-o]-o[-o] 3 =o-o[-o]-o[][][][] 3 You cannot take "-o]-o[" as X' because the brackets are unmatched.

Googologism
Flan Number Fourth Form Revised Twice (F4:2 for short)= o-o[<5> o] 5

Approximation
$$F_{4:2}=N_{\epsilon_{\omega}}(5)$$, where $$N$$ is NGH (No Growing Hierarchy) defined below:

$$N_0(x)=6$$

$$N_{\alpha+1}(x)=N_{\alpha}^x(x)$$

$$N_{\alpha}(x)=N_{\alpha[x]}(x)$$ iff $$\alpha$$ is a limit ordinal.