I have further developed Caret-Star Notation since my last blog post, and am currently developing the notation toward the power of "arrowal" BEAF arrays. I have currently only developed the notation up to about \(\Gamma_0\), but I will update this blog post until I reach \(\varphi(1, 0, 0, 0)\) or the Ackermann ordinal, which is given in the title. In case you haven't read my site's article, here are some of the definitions past order type \(\omega\):
a(*^[0](1)^b)c = a(*^[0]^b)(a(*^[0]^b)(a(*^[0]^b)(...(a(*^[0]^b)(a(*^[0]^b)a))...))) w/ c+1 copies of a
a(*^[0](d)^b)c = a(*^[0](d-1)^b)(a(*^[0](d-1)^b)(a(*^[0](d-1)^b)(...(a(*^[0](d-1)^b)(a(*^[0](d-1)^b)a))...))) w/ c+1 copies of a
a(*^[1]^b)c = a(*^[0](c, 0)^b)a
a(*^[d]^b)c = a(*^[d-1](c, 0, 0, ...(d+1 arguments)... 0, 0)^b)a
a(*^[*]^b)c = a(*^[c]^b)a (This is already on par with Bowers' linear arrays)
a(*^[*, 1]^b)c = a(*^[*](((...(((1)))...)))^b)a w/ c pairs of parentheses around the lone 1 (on par with planar arrays)
For more on the notation up to \(\varepsilon_0\), read my site's article . Note that I have not yet updated some of part 2 (linked at the end of the article) from my original idea.
After reaching \(\varepsilon_0\) at a(*^[[0]]^b)c, we can continue using an approach somewhat similar to how Sbiis Saibian continued Hyper-E beyond this level.
a(*^[[0]](*^^*)^b)c = a(*^[[0]]<<1>>(<1>(<1>(...(<1>(<1>(<1>)))...)))1^b)a w/ c-2 layers if c >= 3, a(*^[[0]]((1))^b)a if c=2, and a(*^[[0]](1, 0)^b)a if c=1 (\(\varepsilon_0 \times 2\))
a(*^[[0]](*^^*+1)^b)c = a(*^[[0]](*^^*)(1)^b)c = a(*^[[0]](*^^*)^b)(a(*^0(*^^*)^b)(a(*^[[0]](*^^*)^b)(...a(*^[[0]](*^^*)^b)(a(*^[[0]](*^^*)^b)a))...))) w/ c+1 copies of a
a(*^[[0]](2(*^^*))^b)c = a(*^[[0]](*^^*)(*^^*)^b)c = a(*^[[0]](*^^*)<<1>>(<1>(<1>(...(<1>(<1>(<1>)))...)))(1)^b)a w/ c-2 layers if \(c \ge 3\) (FGH: \(\varepsilon_0 \times 3\)
a(*^[[0]](*^(*^^*+1))^b)c = a(*^[[0]](c(*^^*))^b)a (\(\omega^{\varepsilon_0 + 1}\))
a(*^[[0], 1]^b)c = a(*^[[0]]({*^^*}^{1})^b)c = a(*^[[0]](*^(*^(*^(...(*^(*^(*^^*+1)))...)))^b)a w/ c copies of *^ (\(\varepsilon_1\))
a(*^[[0], 2]^b)c = a(*^[[0], 1]({*^^*}^{2})^b)c = a(*^[[0], 1](*^(*^(*^(...(*^(*^({*^^*}^{1}+1)))...)))^b)a w/ c copies of *^ (\(\varepsilon_2\))
a(*^[[0], *]^b)c = a(*^[[0], c]^b)a (\(\varepsilon_\omega\))
a(*^[[0, *, 1]^b)c = a(*^[[0], *]({*^^*}^{*^2})^b)c = a(*^[[0], *]({*^^*}^{c*})^b)a (\(\varepsilon_{\omega^2}\))
a(*^[[0(1)]]^b)c = a(*^[[0], *, *, *, ...(c *s)..., *, *, *]^b)a (\(\varepsilon_{\varepsilon_0}\))
a(*^[[0(1)
a(*^[[0], [0], 1]^b)c = a(*^[[0], [0]]({*^^*}^{*^^*}^{1})^b)c = a(*^[[0], [0]]({*^^*}^{*^(*^(*^(...(*^(*^({*^^*}^{1}+1)))...)))})^b)a w/ c layers inside the second pair of braces (\(\varepsilon_{\varepsilon_1}\))
a(*^[[0], [d]]^b)c = a(*^[[0], [d-1], *, *, *, ...(c *s)..., *, *, *]^b)a (\(\varepsilon_{\varepsilon_{\varepsilon_0}}\))
a(*^[[1]]^b)c = a(*^[[0(c)]]^b)a (\(\zeta_0\))
a(*^[[1], 1]^b)
a(*^[[1(0, 1)]]^b)c = a(*^[[1], *, *, *, ...(c *s)... , *, *, *]^b)a (\(\zeta_{\varepsilon_0}\))
a(*^[[1(1, 0)]]^b)c = a(*^[[1(0, c]]^b)a \(\zeta_{\zeta_0}\))
a(*^[[2]]^b)c = a(*^[[1(c, 0)]]^b)a \(\eta_0\)
a(*^[[3]]^b)c = a(*^[[2(c, 0, 0)]]^b)a \(\varphi(4, 0)\)
a(*^[[*]]^b)c = a(*^[[c]]^b)a (\(\varphi(\omega, 0)\))
a(*^^b)c = a(*^[[*]](*^^*^*^2)^b)c = a(*^[[*]](*^^*^(c*))^b)c
a(*^[[*, *]]^b)c = a(*^[[*, c]]^b)a (\(\varphi(\varepsilon_0, 0)\))
more
...
And after all that, we finally reach a(*^[[[0]]]^b)c = a(*^[[*, *, *, ..., *, *, *]]^b)a w/ c *s inside the brackets, which has a growth rate of \(\Gamma_0\), on par with pentational arrays.
We can denote a(*^[[[...[[[0]]]...]]]^b)c w/ n bracket pairs as a(*^{n}^b)c, and define a(*^{*}^b)c as a(*^{c}^b)a. This reaches a growth rate of \(\varphi(\omega, 0, 0)\)), the power of {X, X, X} arrays.
- ^^^ STAY TUNED ^^^*
And finally, we reach a(*^{^{^{^...{^{^{1}^}^}...^}^}^}^b)c, which has a growth rate of \(\varphi(1, 0, 0, 0)\) (the Ackermann ordinal) as the number of bracket pairs increases.