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Non-arithmetical analysis - Printable Version

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Non-arithmetical analysis - Xorter - 04/27/2017

Hello, Everybody!
This is the 800th post in this topic. Well, what if I define non-arithmetical operational derivatives and integrals on the following way. But before it, I would like to say about that we have 9 so-called ancient methods (operations) from arithmetical addition through functional multiplication to logical nand. Here is just a few example for the non-arithmetical derivation:
(N = 1)
It can be iterated or extended to other ancient methods and to integrals (non-arithmetical anti-derivatives).
And yes, as you can see, there is a derivative derivation, too ... interesting, is not it?
My question is how to calculate of these iterated ancient methods?
For example what is f^↑infinity?