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.
 A: 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.
A: Write out the signature of your theory: A list of names of constants, names of functions (and their arity), names of relations and their arity.
Then list the axioms of the theory: logical sentences which must hold.
Then produce some model of your theory: Give some set and an interpretation that translates every name of a constant to an element of the set. Every function of a function on this set.. etc.
That a model exists proves the consistency of your theory (at least, it's consistent if your metatheory is).

Example.
Signature $(T,F,not,and,or)$. Axioms $\forall x, not(not(x))=x$ and $\forall x y, and(T,F)=T$ etc.. Model: {0,1} interpretation $T \mapsto 1$, $F \mapsto 0$, $not \mapsto $ the $1-x$ function, $and \mapsto$ the $xy$ function. This proves the theory of boolean algebra is consistent.
