User blog comment:Googology Noob/Ordinal FGH, with an actual definition!/@comment-1605058-20151220095905

I have just noticed a flaw in your definition, but I'm pretty sure this is just you miswriting what you actually mean:

Consider evaluating $$f_{\omega^2}(\omega)$$ from the basic rules you give:

$$f_{\omega^2}(\omega)=f_{\omega^2[\omega]}(\omega)=f_{\omega^2}(\omega)$$

where the last equality is $$\alpha[\omega]=\alpha$$. This doesn't lead us anywhere, and no other rule can be applied at this point. What you probably mean is for $$f_{\omega^2}(\omega)$$ to be defined as limit of $$f_{\omega^2[n]}(\omega)$$.

While writing this comment I've noticed a far more serious issue: consider computing $$f_{\omega^2}(\omega+1)$$. After one reduction, you get $$f_{\omega^2[\omega+1]}(\omega+1)=f_{\omega^2+\omega}(\omega+1)$$, and now we have to deal with expression with a larger ordinal in index. Further reductions give $$f_{\omega^2+\omega+1}(\omega+1)$$ and then f_{\omega^2+\omega}^{\omega+1}(\omega), so in short - it only gets worse. This cannot be fixed like above.