User blog comment:B1mb0w/Fundamental Sequences/@comment-5529393-20151113121456

Your alternative rule set does not work currently. It is not true that φ ( α , β + 1 )  = φ ( α , β ) ↑ ↑ <span class="mi" id="MathJax-Span-1111" style="font-weight:normal;padding-left:0.263em;vertical-align:0px;transition:none;display:inline;position:static;line-height:normal;font-size:19.2px;white-space:nowrap;font-family:MathJax_Math-italic;">ω ;  instead, we have $$\varphi (\alpha, \beta) \uparrow \uparrow \omega = \varepsilon_{\varphi(\alpha, \beta) + 1}$$, which is not equal to $$\varphi (\alpha, \beta+1)$$ unless $$\alpha = 1$$.

I also take issue with your statements saying that the calculated example is simpler with your ruleset than with the original ruleset. The rules used for $$ \varphi(\alpha + 1, 0)$$ are exactly the same; the only difference is in presentation. For example, you say it is "artificial" to say that $$\varphi(\alpha+1,0)[0] = 0$$, but it is the same with your rule $$\varphi(\alpha+1,0)[n] = \varphi^n(\alpha, 0_*)$$.