= Eventual dominance in a fast-growing hierarchy
{title2=$\alpha<\beta\Longrightarrow F_\alpha(n)<F_\beta(n)\text{ eventually}$}
For the standard <fundamental sequences of limit ordinals> below <epsilon zero>, the index order corresponds to eventual, rather than necessarily pointwise, growth. For example, under $\omega[n]=n+1$ and successor iteration $n+1$ times, $F_\omega(1)=F_2(1)=7$ whereas $F_3(1)=2047$. Limit stages are diagonal choices, so a larger index can give a smaller value at small arguments.
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