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ESMN (Extended Steinhaus-Moser Notation)

Background

Now I have been trying to understand dimensional arrays for an inordinate amount of time. As you'd expect I'm not a smart person. So instead of taking the time to understand them, I took the lazy way out and "created" (probably not) a "new" (again probably not) "system" (there is not much that is system-ish about this at all) of dimensional arrays that work the way I imagined it (it's probably going to be very bad).

Array of function

To diagonalize over the linear arrays we can use an adaptation of Jonathan Bower's array of function.

Rules

1)  i) [n] \(\lambda_{0}\) becomes [n, n, ... n, n] with n number of n'sThis can alternatively be displayed as \(\begin{bmatrix} n \\ 1 \end{bmatrix}\)

ii)Generally \(\begin{bmatrix} n \\ m \end{bmatrix}\) = [[...[[n]\(\lambda_{0}\)]...]\(\lambda_{0}\)]\(\lambda_{0}\) with n number of \(\lambda_{0}\) 's

2) i) Next we can define \({\lambda_{0}}^{2}\) [n] as a 2-dimensional array of n's or \(\begin{bmatrix} n & n &\cdots n \\ n & n &\cdots n \\ \vdots & \vdots & \ddots & \vdots \\n & n &\cdots n \end{bmatrix}\) with n number of rows each row will be evaulated using rules 1)i) and ii). To evaulate sequences of numbers first convert them into linear arrays and evaluate them using the previous ESMN rules.

ii) Similarly like BEAF we can define \({\lambda_{0}}^{n}\) [n] as a n-dimensional hypercube of linear arrays with all the entries being n.

From this we can define \(\lambda_{0} \uparrow \lambda_{0}\) [n] = \(\lambda_{0} \uparrow n\), \(\lambda_{0} \uparrow \uparrow \lambda_{0}\) [n] = \(\lambda_{0} \uparrow \uparrow n\) and \(\lambda_{0} \{m\} \lambda_{0} \) [n] as \(\lambda_{0} \{m\} n\).

The "limit" of this notation is \(\lambda_{0}\{\lambda_{0}\}\lambda_{0}\). If these \(lambda_{0}\) functions work like I think they do, this should be decently large, even by amateur googologist's standards. If my analysis is correct and if the \(\lambda_{0}\) is simillar to the ordinal \(\omega\), the "limit" of this system (again not many system-ish things about this) would possess a growth rate of \(\Gamma_{0}\). Feel free to critizise and ask about anything in this blog post. I realise that this is very rushed and most probably (I mean definitely) contain many errors. Feedback is always welcome and much appreciated (so long as it is constructive). Thank you for taking the time to read (and analyze if you did) this.


Again thanks.





Thanks.





I said thank you.





You can leave now.





Hey, I said you can leave now.





Whay are ou still reading this? Don't you have better things to do?





Stop reading this.





Stop it.





Right now.





Good god you actually took the time to scroll just to look for more things?





Seriously? You keep scrolling?





Is your mouse or trackpad broken? Why are you reading this?





For the love of god or any deity you choose to worship stop scrolling.















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You are a curious little thing aren't you. Fine since you took the time to scroll all the way down here I'll share a little tidbit with you. Did you notice something something funny with the \(\lambda_{0}\) ? Something seemingly unnecessary, hmm? Perhaps a seemingly trivial subesction that always is looked down upon? : )

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