Hyperfactorial array notation: Analysis part 3

This has been corrected for new definitions. 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.

One thing I have noticed lately which is technically a mistake in my definitions is that in my notation, \(\Gamma_1\) should be [1]w/[1]w/[1]w/....w/[1]w/[2(1)2]. In my notation, the Gamma 1 equivalent, [1(1)3], is actually [1(1)2]w/[1(1)2]w/...w/[1(1)2], and therefore strangely enough, the mistake was actually that my notation is more powerful than expected!

If unspecified, assume a ... means omega+1 (+1 just so it's not a limit ordinal) repeats.

Type-2 brackets

Hyperfactorial array (without the n!)	FGH ordinal
[1(1)2]	$\Gamma_0=\psi(\Omega^\Omega)$
[1(1)k] (is actually greater than (see 2nd paragraph at top))	$\Gamma _{k-2}=\psi (\Omega ^{\Omega }\times (k-1))$
[1(1)[1(1)2]]	$\Gamma _{\Gamma _{0}}=\psi (\Omega ^{\Omega }\times \psi (\Omega ^{\Omega }))$
[1(1)[21]] = [1(1)[1(1)[1(1)[...[1(1)n]...]]]]	$\psi (\Omega ^{\Omega }\times \Omega )$
[1(1)[21]]w/[1] = [1,1,...,1,2(1)[21]]	$\phi (\omega ,\psi (\Omega ^{\Omega }\times \Omega ))=\psi (\Omega ^{\omega }\times \psi (\Omega ^{\Omega }\times \Omega ))$
[1(1)[22]] = [1(1)[21]]w/[1(1)[21]]w/...w/[1(1)[21]] > [1]w/[1]w/[1]w/....w/[1]w/[2(1)[21]] = next Gamma after [1(1)[21]]	$\psi (\Omega ^{\Omega }\times (\Omega +1))$
[1(1)[2k]] = [1(1)[2k-1]]w/[1(1)[2k-1]]w/...w/[1(1)[2k-1]] > [1]w/[1]w/[1]w/....w/[1]w/[2(1)[2k-1]] = next Gamma after [1(1)[2k-1]]	$\psi (\Omega ^{\Omega }\times (\Omega +(k-1)))$
[1(1)[21,2]] = [1(1)[2[21]]]	$\psi (\Omega ^{\Omega }\times (\Omega +\Omega ))=\psi (\Omega ^{\Omega }\times (\Omega \times 2))$
[1(1)[22,2]] > [1]w/[1]w/[1]w/....w/[1]w/[2(1)[21,2]] = next Gamma after [1(1)[21,2]]	$\psi (\Omega ^{\Omega }\times (\Omega \times 2+1))$
[1(1)[2k,2]] > [1]w/[1]w/[1]w/....w/[1]w/[2(1)[2k-1,2]] = next Gamma after [1(1)[2k-1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega \times 2+(k-1)))$
[1(1)[21,3]] = [1(1)[2[21],2]]	$\psi (\Omega ^{\Omega }\times (\Omega \times 2+\Omega ))=\psi (\Omega ^{\Omega }\times (\Omega \times 3))$
[1(1)[21,4]] = [1(1)[2[21],3]]	$\psi (\Omega ^{\Omega }\times (\Omega \times 4))$
[1(1)[21,k]] = [1(1)[2[21],k-1]]	$\psi (\Omega ^{\Omega }\times (\Omega \times k))$
[1(1)[21,1,2]] = [1(1)[21,[21]]]	$\psi (\Omega ^{\Omega }\times (\Omega \times \Omega ))=\psi (\Omega ^{\Omega +2})$
[1(1)[2k,1,2]] = (k-1)th gamma after [1(1)[21,1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{2}+(k-1)))$
[1(1)[21,2,2]] = [1(1)[2[21,1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{2}\times 2))$
[1(1)[21,k,2]] = [1(1)[2[21,1,2],k-1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{2}\times k)$
[1(1)[21,1,3]] = [1(1)[21,[21],2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{2}\times \Omega ))=\psi (\Omega ^{\Omega }\times (\Omega ^{3}))$
[1(1)[21,1,4]] = [1(1)[21,[21],3]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{3}\times \Omega ))=\psi (\Omega ^{\Omega }\times (\Omega ^{4}))$
[1(1)[21,1,k]] = [1(1)[21,[21],k-1]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{(}k-1)\times \Omega ))=\psi (\Omega ^{\Omega }\times (\Omega ^{k}))$
[1(1)[2a,b,c]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{c}\times b+a))$
[1(1)[21,1,1,2]] = [1(1)[21,1,[21]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }))$
[1(1)[2k,1,1,2]] = (k-1)th gamma after [1(1)[21,1,1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }+(k-1)))$
[1(1)[2[21],1,1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }+\Omega ))$
[1(1)[21,2,1,2]] = [1(1)[2[21,1,1,2],1,1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }\times 2))$
[1(1)[21,3,1,2]] = [1(1)[2[21,1,1,2],2,1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }\times 3))$
[1(1)[21,k,1,2]] = [1(1)[2[21,1,1,2],k-1,1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }\times (k-1)))$
[1(1)[21,[21],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega }\times \Omega ))=\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +1}))$
[1(1)[21,[22],1,2]] = [1(1)[2[21,[21],1,2],[21],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +1}\times 2))$
[1(1)[21,[2k],1,2]] = [1(1)[2[21,[21],1,2],[2k-1],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +1}\times k))$
[1(1)[21,[21,2],1,2]] = [1(1)[21,[2[21]],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +2}))$
[1(1)[21,[22,2],1,2]] = [1(1)[2[21,[21,2],1,2],[21,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +2}\times 2))$
[1(1)[21,[21,3],1,2]] = [1(1)[21,[2[21],2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +3}))$
[1(1)[21,[21,k],1,2]] = [1(1)[21,[2[21],k-1],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega +k}))$
[1(1)[21,[21,1,2],1,2]] = [1(1)[21,[21,[21]]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 2}))$
[1(1)[21,[2k,1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 2}\times (k-1)))$
[1(1)[21,[2[21],1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 2+1}))$
[1(1)[21,[2[21,2],1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 2+2}))$
[1(1)[21,[2[21,k],1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 2+k}))$
[1(1)[21,[21,2,2],1,2]] = [1(1)[21,[2[21,1,2],1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 3}))$
[1(1)[21,[2[21],2,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 3+1}))$
[1(1)[21,[2[21,2],2,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 3+2}))$
[1(1)[21,[21,3,2],1,2]] = [1(1)[21,[2[21,1,2],2,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times 4}))$
[1(1)[21,[21,k,2],1,2]] = [1(1)[21,[2[21,1,2],k-1,2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega \times k}))$
[1(1)[21,[21,1,3],1,2]] = [1(1)[21,[21,[21],2],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}}))$
[1(1)[21,[2[21],1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}+1}))$
[1(1)[21,[2[21,2],1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}+2}))$
[1(1)[21,[2[21,1,2],1,3],1,2]] = [1(1)[21,[2[21,[21]],1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}+\Omega }))$
[1(1)[21,[2[21,2,2],1,3],1,2]] = [1(1)[21,[2[2[21,1,2],1,2],1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}+\Omega \times 2}))$
[1(1)[21,[2[21,k,2],1,3],1,2]] = [1(1)[21,[2[2[21,1,2],k-1,2],1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}+\Omega \times k}))$
[1(1)[21,2,3],1,2]] = [1(1)[21,[2[21,1,3],1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}\times 2}))$
[1(1)[21,k,3],1,2]] = [1(1)[21,[2[21,1,3],k-1,3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{2}\times k}))$
[1(1)[21,1,4],1,2]] = [1(1)[21,[21,[21],3],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{3}}))$
[1(1)[21,1,k],1,2]] = [1(1)[21,[21,[21],k-1],1,2]]	$\psi (\Omega ^{\Omega }\times (\Omega ^{\Omega ^{k-1}}))$
[1(1)[21,1,2,2]] = [1(1)[21,[21,1,1,2],1,2]]	$\psi (\Omega ^{\Omega }\times \Omega ^{\Omega ^{\Omega }})\approx \psi (\Omega ^{\Omega ^{\Omega }})$

It should be obvious by now that Omega brackets here work in exactly the same way as the ones here, with the only difference being that the brackets analysed by Cloudy176 (through the link) use lower case omega, and these upper case omega, I believe that the evidence from the examples above all exactly matching Cloudy's work (if they don't they probably should: point out anything that looks unusual), should be enough to just skip the rest of the analysis contained on the 1st analysis page, to save typing out pretty much the same thing exactly again and just take a few of the more important comparisons:

Continued Type-2 Brackets

Hyperfactorial array (without the n!)	FGH ordinal
[1(1)[21,1,1,1,2]]	$\psi (\varepsilon _{\Omega +1})=\psi _{0}(\psi _{1}(0))$
[1(1)[21,1,1,1,1,2]]	$\psi (\zeta _{\Omega +1})=\psi _{0}(\psi _{1}(\Omega _{2}))$
[1(1)[21(1)2]]	$\psi (\Gamma _{\Omega +1})=\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}}))$
[1(1)[21(1)1(1)2]]	$\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}}\times 2))$
[1(1)[21([1])2]]	$\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}^{\Omega _{2}}}))$

...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. Type-3 brackets will work in exactly the same way as type-2 brackets except that they will nest type-2 brackets rather than type 1. Because the rules are exactly the same in every respect apart from this, there really is no point copying out the above comparisons again. (if you really want them, just look at the stuff above, and increase all the subscripts on brackets and Omegas by 1.

Up to \(\psi_0(\varepsilon_{\Omega_{\omega}+1})\): the TFB ordinal

Hyperfactorial array (without the n!)	FGH ordinal
[1(1)[21(1)k]]	$\psi (\Gamma _{\Omega +(k-1)})=\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}}\times k))$
[1(1)[21(1)[21(1)2]]]	$\psi (\Gamma _{\Gamma _{\Omega +1}})=\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}}\times \psi _{1}(\Omega _{2}^{\Omega _{2}})))$
[1(1)[21(1)[31]]]	$\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}}\times \Omega _{2}))$
[1(1)[21(1)[31,1,2]]]	$\psi _{0}(\psi _{1}(\Omega _{2}^{\Omega _{2}+2}))$
[1(1)[21(1)[31,1,1,1,2]]]	$\psi _{0}(\psi _{1}(\varepsilon _{\Omega _{2}+1}))=\psi _{0}(\psi _{1}(\psi _{2}(0)))$
[1(1)[21(1)[31(1)2]]]	$\psi _{0}(\psi _{1}(\psi _{2}(\Omega _{3}^{\Omega _{3}})))$
[1(1)[21(1)[31([1])2]]]	$\psi _{0}(\psi _{1}(\psi _{2}(\Omega _{3}^{\Omega _{3}^{\Omega _{3}}})))$
[1(1)[21(1)[31(1)[41(1)2]]]]	$\psi _{0}(\psi _{1}(\psi _{2}(\psi _{3}(\Omega _{4}^{\Omega _{4}}))))$
[1(1)[[1]1([1])2]]	$\psi _{0}(\Omega _{\omega }^{\Omega _{\omega }^{\Omega _{\omega }}})$

All the full definitions (going to be at some point: they still need a little work) are on my website here (including fixed ones mentioned in the comments in the last post).