User blog:Ikosarakt1/Introduction to pentational arrays

In this blog post I'll introduce my vision to how pentational arrays work. I will start with structures only a bit larger that tetrational, then, when the main gist will be explained, I'll go to pentationals and probably even beyond.

First, remember my blog post: formal definition of tetrational arrays. There I proposed to enclose structures into separators: \(A \&\ b[p] = \{b,p (A) 2\}\), where A is the any structure. It allows to make it more natural rather than use arrays in arrays.

So, what is the \((X \uparrow\uparrow X)\) separator? I decided to use Bowers' ideas and used curly brackets. Let \((X \uparrow\uparrow X) = \{X\} = \{X^X\} = \{X^{X^X}\} = \{X^{X^{X^X}}}\} = \{\underbrace{X^{X^{\cdots^{X^X}}}}}_{n\text{ }X's\}\) In otherwords, there is no matter how high the power tower enclosed in curly brackets, because X can represent arbitrary number of them. To distinguish it, however, from the other similar separators in the form \((X \uparrow\uparrow n)\), I've used curly brackets. So, \((\{X^X\}) = (X \uparrow\uparrow X\), but \((X^X) = (X \uparrow\uparrow 2)\). When the array in the form \{b,p (\{X^{X^{\cdots^{X^X}}}\}) 2\}, it is equal to \{X \uparrow\uparrow X \&\ b[p]\}

This is the sketch of the new rule for my pentational arrays. All rules for tetrational arrays should be applied ignoring curly brackets.

For example:

\(\{3,3 (\{X^{X+1}\}\) 2\} = \{3,3 (\{X^X\}*X\}) 2\} = \{3,3 (\{X^X\}*3\}) 2\} = \{3,3 (\{X^X\}+\{X^X\}+\{X^X\}) 2\} = \{\{X^X\} \&\ 3 ({X^X}) \{X^X\} \&\ 3 ({X^X}) \{X^X\} \&\ 3\} = \{X \uparrow\uparrow X \&\ 3 ({X^X}) X \uparrow\uparrow X \&\ 3 ({X^X}) X \uparrow\uparrow X \&\ 3\}. Imagine that when it, eventually, after really long time, becomes \{3,N ({X^X}) \#\}, we should construct \(X \uparrow\uparrow N \&\ 3\), and N is very, very large. And we're barely scratch the surface of pentational arrays.

The next natural question is: what is \(X \uparrow\uparrow (X+1)\) structure? Well, we can think about \(X+1\) as a row with X items (arbitrary number of them), and 1 item above (or below) it. Thus, we can use curly brackets as a row delimiters and \((X \uparrow\uparrow (X+1)) = ({\{X^{X^{\cdots^{X^X}}}\}}^X)\). That is, we can still apply the same rules (including my new, that I don't defined formally, but I believe that you understood how it work).

For example:

\(X \uparrow\uparrow (X+1) \&\ 3[2] = \{3,2 ({\{X^X\}}^X) 2\} = \{3,2 ({{\{X^X\}}^2) 2\} = \{3,2 (\{X^{X*X}\}) 2\} = \{3,2 (\{X^{X*2}\}) 2\} = \{3,2 (\{X^{X+X}\}) 2\} = \{3,2 (\{X^{X+2}\}) 2\} = \{\{X^{X+1}\} \&\ 3[2] (\{X^{X+2}\}) \{X^{X+1}\} \&\ 3[2]\} = \{3,3 (X) 3,3 (\{X^X\}) 3,3 (X) 3,3 (\{X^{X+1}\}) 3,3 (X) 3,3 (\{X^X\}) 3,3 (X) 3,3\}\).

Using my rules, we can notate and solve the further structures:

\(X \uparrow\uparrow (X+2) = {{\{X^{X^{\cdots^{X^X}}}\}}^X}^X}\)

\(X \uparrow\uparrow (X+3) = {{{\{X^{X^{\cdots^{X^X}}}\}}^X}^X}^X\)

\(X \uparrow\uparrow (X*2) = {{\{X^{X^{\cdots^{X^X}}}\}}^{{\{X^{X^{\cdots^{X^X}}}\}}\)

\(X \uparrow\uparrow (X*2+1) = {{\{X^{X^{\cdots^{X^X}}}\}}^{{{\{X^{X^{\cdots^{X^X}}}\}}^X}\)

\(X \uparrow\uparrow (X*3) = {{\{X^{X^{\cdots^{X^X}}}\}}^{{{\{X^{X^{\cdots^{X^X}}}\}}^{{\{X^{X^{\cdots^{X^X}}}\}}}\)

Now I really need to introduce some new type of separator that separates planes of X's: let normal curly brackets \(\{\}\) become \(\{\}^X\), and the new type should be \(\{\}^{X^2}\). Now \(X \uparrow\uparrow (X^2) = \{\cdots\cdots\}^{X^2}\), where \(\cdots\cdots\) just means \(X \times X\) grid of X's, X sections of X's separated by the normal curly brackets.

For example:

\(X \uparrow\uparrow X^2 \&\ 3[2] = \{3,2 ({\{X^X\}}^{X^X}) 2\} = \{3,2 ({\{X^X\}}^{X^2}\}) 2\} = \{3,2 ({\{X^X\}}^{X*X}\}) 2\} = \{3,2 ({\{X^X\}}^{X*2}\}) 2\}\), etc.

Then we can make \(\{\}^{X^3}\)-typed brackets that separates realms, \(\{\}^{X^4}\)-typed brackets that separates flunes, and so on. Then you may ask, how to transform \(X \uparrow\uparrow A\) structure into a separator? Below is the process that shows how to convert, \(X \uparrow\uparrow X^2 \&\ 3[2]\) into \(\{3,2 ({\{X^X\}}^{X^X}) 2\}:

\(X \uparrow\uparrow X^2 \&\ 3[2] = X \uparrow\uparrow (X*X) \&\ 3[2] = X \uparrow\uparrow (X*2) \&\ 3[2] = X \uparrow\uparrow (X+X) \&\ 3[2] = X \uparrow\uparrow (X+2) \&\ 3[2] = X^{X \uparrow\uparrow (X+1)} \&\ 3[2] = {\{X^X\}}^{X \uparrow\uparrow X} \&\ 3[2] = {\{X^X\}}^{X \uparrow\uparrow 2} \&\ 3[2] = {\{X^X\}}^{X^X} \&\ 3[2] = \{3,2 ({\{X^X\}}^{X^X}) 2\}.

In other words, when A is a successor structure, we take \(X \uparrow\uparrow A \&\ b[p] = X^{X \uparrow\uparrow (A-1)} \&\ b[p]\), if it is form \(X \uparrow\uparrow (B+C) \&\ b[p]\), we take entire part of power tower below it into C-typed brackets, \(\{\}^C\).

Notice that C can be also tetrational structure itself, for instance \(X \uparrow\uparrow (X \uparrow\uparrow (X+1))\), and we eventually can come to the point where one type of brackets expresses another. This is the limit of pentational arrays.