User blog comment:B1mb0w/Rule 2B/@comment-27513631-20160206233504

$$\varepsilon_\gamma=\gamma\Rightarrow\gamma\uparrow\uparrow\omega = \varphi(1,\gamma+1)=\varesilon_{\gamma+1}$$, for practically all definitions of $$\varphi$$ and $$\uparrow\uparrow\omega$$.

Also, would you care to emlighten me as to how you can make ordinals like $$\zeta_0$$ and $$\omega_1$$ using $$\uparrow\uparrow\omega$$?

If you're using the normal $$\varphi$$ function, then the supremum of the values in most of your fundamental sequences for ordinals produce ordinals lower than the ones you were taking the fundamental sequences for.