User blog:Edwin Shade/Defining LOCC (Language Of Ordinal Construction)

LOOC is a language that I personally have created to describe the construction of trans-infinite ordinals, it is composed of the following operations, (which will be subject to change as I develop it):

$${\phi}$$ denotes the first fixed point of the function $$\phi$$, or the first point at which the function's input is the same as it's output.

$$:$$ is a symbol to be used when defining an ordinal, for instance, $$L:{\phi}$$ means "the ordinal L is defined as the fixed point of the function $$\phi$$.

$$\mapto$$ is a symbol to be used when separating a function's input from it's output. So for instance, $$n\mapsto n^n$$ means, "a function that takes an input n and outputs $$n^n$$.

$$_$$ is just an underscore, and gives one the potential to create recursive equations in LOOC, such as Z_(n+1)=(Z_n)^2+1.

$$+$$ denotes addition.

$$*$$ denotes multiplication.

$$^$$ denotes exponentiation.

$$^^$$ denotes tetration, and although it is shown as two symbols here it is to be regarded as one.

$$(\phi)$$ indicates that $$\phi$$ is an expression isolated from terms that may be outside of the parenthesis. Parenthesis in LOCC act just as they do in regular algebra.

By using combinations of the above operations we may define such ordinals as $$\epsilon_0$$, which would be {b$$\mapsto$${c$$\mapsto$$c+1}^b}. This would be read as "the fixed point of the function 'b maps to "the fixed point of c maps to c plus 1"' to the power of b".

For those wondering how fundamental sequences are defined, it is simple. Fundamental sequences are contained within the definition of the ordinal's fixed point. For an ordinal whose fixed point is defined as {a$$\mapsto f(a)}$$, the nth element of it's fundamental sequence is f(n), where n begins at 0 and proceeds up with all the natural numbers.

LOCC is to be used in conjunction with NAN, (Nova Array Notation), described in this blog post.

If there are any questions, comments, or examples needed I will provide them. Note that LOCC is by no means finished, I anticipate it will be however in a few days.