User blog comment:B1mb0w/FGH of Omega/@comment-27513631-20160608172607/@comment-27513631-20160617112950

$$\varphi(\alpha,\beta)=\min\{\gamma:(\forall\delta<\gamma)(\delta+\gamma=\delta) \\\wedge(\forall\delta<\alpha)(\varphi(\delta,\gamma)=\gamma)\wedge(\forall\delta<\beta)(\varphi(\alpha,\delta)<\gamma)\}$$ Simple! (Assuming you meant binary and not transfinitely-indexed Veblen).

Also, define: $$\\ \text{deg}(0)=0 \\ \alpha<\varphi(\alpha,\beta)\wedge\gamma<\omega\cdot\varphi(\alpha,\beta)\Rightarrow \\ \text{deg}(\varphi(\alpha,\beta)+\gamma) = \max\{\text{deg}(\alpha),\text{deg}(\beta),\text{deg}(\gamma)\}+1$$ and $$\\ \text{F}(n)=n \\ \alpha\geq\omega\Rightarrow\text{F}(\alpha)=\text{F}(\max\{\beta<\alpha:\text{deg}(\beta)=2\cdot\text{deg}(\alpha)\})$$

$$\text{F}((\alpha\mapsto\varphi(\alpha,0))^n(0))$$ is a $$\Gamma_0$$-level function of n, defined in 5 lines (with two of those as base cases).