Absolute Omegaggol

Let \(z\) be an empty string or a string consisting of one or more comma-separated zeros \(0,0,...,0\) and \(s\) be an empty string or a string consisting of one or more comma-separated ordinals \(\alpha _{1},\alpha _{2},...,\alpha _{n}\) with \(\alpha _{1}>0\). The binary function \(\varphi (\beta ,\gamma )\) can be written as \(\varphi (s,\beta ,z,\gamma )\) where both \(s\) and \(z\) are empty strings.
The finitary Veblen functions are defined as follows:
\(\varphi (\gamma )=\omega ^{\gamma }\)
\(\varphi (z,s,\gamma )=\varphi (s,\gamma )\)
if \(\beta >0\), then \(\varphi (s,\beta ,z,\gamma )\) denotes the \((1+\gamma )\)-th common fixed point of the functions \(\xi \mapsto \varphi (s,\delta ,\xi ,z)\) for each \(\delta <\beta\)

Normal Form for finitary Veblen function

We simultaneously define: predicate \(_{FNF}\) (normal form for finitary Veblen function);
set \(T_F\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only the symbols \(0, +,\varphi\) where \(\varphi\) denotes finitary Veblen function;
set \(P_F\) (subset of \(T_F\) which includes only additive principal numbers).
Definition of \(_{FNF}\)
\(\alpha=_{FNF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_F\)
\(\alpha=_{FNF}\varphi(\alpha_1,...,\alpha_n)\) iff \( \alpha=\varphi(\alpha_1,...,\alpha_n)\wedge \alpha_1,...,\alpha_n<\alpha\wedge0<\alpha_1\)
Definition of sets \(T_F\) and \(P_F\)
\(P_F \subset T_F\)
\(0 \in T_F\)
If \(\alpha=_{FNF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots , \alpha _{n}\in P_F\) then \(\alpha \in T_F\)
If \(\alpha=_{FNF}\varphi(\alpha_1,...,\alpha_n)\) and \( \alpha_1,...,\alpha_n\in T_F\) then \(\alpha \in P_F\)

Fundamental sequences

If \(\alpha=\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})\) then \(\alpha[n]=\varphi (s_{1})+\varphi (s_{2})+\cdots +(\varphi (s_{k})[n])\)
If \(\alpha=\varphi (\gamma )\) then \(\alpha[n]=\left\{{\begin{array}{lcr}n\quad {\text{if}}\quad \gamma =1\\\varphi (\gamma -1)\times n\quad {\text{if}}\quad \gamma \quad {\text{is a successor ordinal}}\\\varphi (\gamma [n])\quad {\text{if}}\quad \gamma \quad {\text{is a limit ordinal}}\\\end{array}}\right.\)
If \(\alpha=\varphi (s,\beta ,z,\gamma )\) then: \(\alpha[0]=0\) and \(\alpha[n+1]=\varphi (s,\beta -1,\alpha[n],z)\) if \(\gamma =0\) and \(\beta\) is a successor ordinal; \(\alpha[0]=\varphi (s,\beta ,z,\gamma -1)+1\) and \(\alpha[n+1]=\varphi (s,\beta -1,\alpha[n],z)\) if \(\gamma\) and \(\beta\) are successor ordinals;
\(\alpha[n]=\varphi (s,\beta ,z,\gamma [n])\) if \(\gamma\) is a limit ordinal;
\(\alpha[n]=\varphi (s,\beta [n],z,\gamma )\) if \(\gamma =0\) and \(\beta\) is a limit ordinal;
\(\alpha[n]=\varphi (s,\beta [n],\varphi (s,\beta ,z,\gamma -1)+1,z)\) if \(\gamma\) is a successor ordinal and \(\beta\) is a limit ordinal.
Next, define \(f_\alpha(n)\) in the FGH using the above fundamnetal sequences as \(\alpha(n)\).
The Omegaggol is
 \(\left. \begin{matrix} &&\varphi(10^{100}\underbrace{0,\cdots 0,})(10^{100})\\ & & \varphi(10^{100}\underbrace{0,\cdots 0,})(10^{100}) \\ & & \;\;\underbrace{\quad\;\; \vdots \quad \;\;}\\ & & \varphi(10^{100}\underbrace{0,\cdots 0,})(10^{100}) \\ & & 10^{100}\quad 0's \end{matrix} \right \} \varphi(10^{100},\underbrace{0,0,\cdots,0,0}_{10^{100}})\)