User blog:Tetramur/The new formalization of BEAF

The recursive set
Firstly, I would like to define the recursive set of all valid expressions.

If one function has slower growth rate than second, first function is inferior, and the other one is superior.

The set of functions where each following function is inferior to its predecessor, is called a descending set of functions.

Basic expressions are defined using arrows or linear array notation on X, so X^3, X^^7, {X,X,3}, {X,X,1,2} are all basic expressions.

The level expressions are defined using inferior function to superior, so X^(X^^X) is level expression with X^^X, but (X^^X)^X is not. X^^{X,X,1,2} is level expression with {X,X,1,2} but {X,X,1,2}^^X is not. (to be fixed)

So, expressions are obtained by using the set of all level expression of basic expressions with descending set of functions. For example, {{X,X,1,2},X,X*2}^^^^(X^^^X^7) is valid expression.

(to be expanded and finished)