User blog:DrCeasium/Hyperfactorial array notation: Analysis part 3

This post will look at extended hyperfactorial array notation. It is simply definied as [k@1[k+11]@2] = [k@1[k@1[k...[k@1[k@1@2]...]@2]@2] with n nests. This is surprisingly powerful. As a small sub rule, if the type-k brackets don't exist, put them in around the type-k+1. This was designed to reach the TFB ordinal, and the type-k brackets work in pretty much the same way to \(\Omega_k\). In the following comparisons I have put the type-k brackets in the second row just to kick start it a bit. Quite a lot of the evaluation relies on the type-k brackets doing to \(\Omega_k\) exactly as type-1 brackets do to \(\omega\). See the type-1 bracket comparisons here and here

Type-2 brackets
...Quite powerful then, but that's nothing compared to what can be done by putting type-3 brackets in type-2 brackets in type-1 brackets and so on: