Omegaggol

Omegaggol is equal to \(f_{b(10^{100},10^{100})}(10^{100})\), with the definition of the b function described below. It was coined by a googology wiki user Mumuji.
Unfortunately, the number is ill-defined as we will explain in "Issues" section.
== Definition ==
b(n,m) is defined recursively as:
* \(b(n,m) = \Gamma_{m}\) if \(n = 0\)
* \(b(n,m) = b(n-1,b(n-1,b(n-1,b(n-1,m))))\) if \(n > 0\)
where \(\Gamma\) is "the gamma in gamma-naught ordinal". Although it is not clarified in the defintion, (n,m) is perhaps intended to be a pair of a non-negative integer and an ordinal.
The omegaggol is defined as \(f_{b(10^{100},10^{100})}(10^{100})\) using the fast-growing hierarchy.
== Issues ==
The fast-growing hierarchy is ill-defined unless we fix a system of fundamental sequences. Since the creator does not fix a single definition of a system of fundamental sequences, it is ill-defined. The same issue has been pointed out for Cheese numbers by the creator.
== The number of the gammas in the defining expression ==
We estimate the number of \(\Gamma\)s in the defining expression \(f_{b(10^{100},10^{100})}(10^{100})\) of omegaggol.
For a non-negative integer \(n\) and an ordinal \(m\), we define an ordinal \(\beta(n,m)\) in the following inductive way:
# If \(n = 0\), then \(\beta(n,m) = m\).
# If \(n > 0\), then \(\beta(n,m) = \Gamma_{\beta(n-1,m)}\).
Roughly speaking, it is "the \(n\)-th iteration" of \(\alpha \mapsto \Gamma_{\alpha}\) applied to \(m\).
For any ordinal \(m\), we have
\begin{eqnarray*}
b(0,m) & = & \Gamma_m = \beta(1,m) \\
b(1,m) & = & b(0,b(0,b(0,b(0,m)))) = \beta(1,\beta(1,\beta(1,\beta(1,m)))) = \beta(4,m) \\
b(2,m) & = & b(1,b(1,b(1,b(1,m)))) = \beta(4,\beta(4,\beta(4,\beta(4,m)))) = \beta(4^2,m)
\end{eqnarray*}
In general, for any non-negative integer \(n\) and any ordinal \(m\), it is easy to see \(b(n,m) = \beta(4^n,m)\) by induction on \(n\).
; Proof.
: If \(n = 0\), then we have shown \(b(n,m) = \beta(1,m) = \beta(4^n,m)\) for any ordinal \(m\). Suppose \(n > 0\). By induction hypothesis, we have \(b(n-1,m) = \beta(4^{n-1},m)\) for any ordinal \(m\). Then we obtain
\begin{eqnarray*}
b(n,m) & = & b(n-1,b(n-1,b(n-1,b(n-1,m)))) \\
& = & \beta(4^{n-1},\beta(4^{n-1},\beta(4^{n-1},\beta(4^{n-1},m)))) \\
& = & \beta(4^n,m)
\end{eqnarray*}
: for any ordinal \(m\), which was what we wanted. □
Therefore the number of \(\Gamma\)s in the expression \(f_{b(10^{100},10^{100})}(10^{100}) = f_{\beta(4^{10^{100}},10^{100})}(10^{100})\) is \(4^{10^{100}}\).
== Approximations in other notations ==
Since the number is ill-defined and the intended value of b function is different from its actual value, we give an approximation of the intended value using the fast-growing hierarchy with respect to an unspecified system of fundamental sequences.
==Sources==
Category:Numbers
Category:Incomplete
Category:Numbers and functions by Mumuji
Category:Bachmann's collapsing level