User blog:Allam948736/Extension of CSN up to Ackermann Ordinal

I have further developed Caret-Star Notation since my last blog post, and reached the power of "arrowal" BEAF arrays. 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 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], [0^b)c = a(*^[[0], *, *, *, ...(c *s)..., *, *, *]^b)a (\(\varepsilon_{\varepsilon_0}\))

a(*^[[0], [0]

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 (\(\zeta_0\))

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.