Oracle I is a number defined using a specially defined function in Diagonalizable @ Notation[1]. The name was coined by AlexJN. It is ill-defined, as is explained in #Issues.
Definition[]
- f(0)=10
- For f(n) of n>0:
- Define a function p(n) which returns the inners of an expression, for example p(3(4)(2)2)=(4)(2)
- f(n)=10(n)p(f(n-1))10
- Oracle I is defined as f(10100).
It can informally be written as 10(10100)(10100-1)(10100-2)...(3)(2)(1)10
Issues[]
The function p with the unspecified domain is unformalised because of the informal identification of a number n and its non-unique expression.
As a consequence, the number is ill-defined.
Approximations[]
Since the number is ill-defined, the following just shows the unsourced expectation by the creator himself.[2]
| Notation | Approximation |
|---|---|
| Bowers' Exploding Array Function | {10,{10,100}(1)2} |
| Bird's array notation | {10,{10,100}[2]2} |
| Cascading-E notation | E10#^#E100 |
| Fast-growing hierarchy | \(f_{\omega^\omega}(f_2(323))\) |
| Hardy hierarchy | \(H_{\omega^{\omega^\omega}}(H_{\omega^2}(323))\) |
| Slow-growing hierarchy | \(g_{\psi(\Omega^{\Omega^\omega})}(g_{\omega^{\omega^2}}(10))\) |