0 - Successor Function
Hierarchal-Structor Notation begins with The Successor Function, which it denotes as |n|.
Let n be a natural number or a positive integer.
If this is so |n| does not denote the absolute value of n, but rather the successor of n.
That is:
|n| = S(n) = n+1
In this context the vertical bars are called Containers.
If the containers contain no number, then the value is the successor of the least allowed input. That is | | = 1 if natural numbers are used for n, and | | = 2 if positive integers are used for n.
In the original formulation of #[] (Hierarchal-Structor Notation) the preference was towards positive integers.
The Singular Successor Function, can be extended to any arity. If every argument other than the 0th argument, is equal to 1 then the result is equivalent to the successor function.
That is:
S(n) = |n| = |n.1| = |n.1.1| = |n.1.1.1| = |n.1.1.1.1| = |n.1.1.1.1.1| = |n.1.1.1.1.1.1| = ...
If the natural numbers are used instead then:
S(n) = |n| = |n.0| = |n.0.0| = |n.0.0.0| = |n.0.0.0.0| = |n.0.0.0.0.0| = |n.0.0.0.0.0.0| = ...
Note that a dot is used to separate arguments instead of a comma. The dot is an example of what is called a delimiter.
The string of alternating arguments and delimiters, wrapped in a container, is what is known as a block. When the containers are removed the string of alternating arguments and delimiters is what is known as a struct, an instantiation of the more general concept of a structor.
The Strength of this "notation", is of order-type 0 by definition, as The Successor Function is defined as the simplest googological function.
Copyright Sbiis Saibian 2025