Mathematically correct definition of own calculus I'm looking for an example of a complete definition of a custom calculus I could learn from to mathematically correct define my own calculus consinsting of:


*

*objects: a, aa, aaa, aaaa, aaaa, ... , b, bb, bbb, bbbb, etc

*operations on those objects: + , - , *; eg: a+a = aa

*etc...


Basically I want to remap natural numbers and the basic functions for my own needs and am looking for good resource to define a mathematically sound framework.

If I understood your question correctly you want to treat natural numbers as strings and you are thinking of typographical theory of natural numbers. These links might turn out helpfull [1], [2]. Second source provides an example of something you have had in mind I guess. In particular, one example of typographical system of natural numbers is presented in Goedel, Escher, Bach" An Eternal Golden Braid - the system is called TNT.
