User blog:Edwin Shade/Revisiting LOOC (Language Of Ordinal Construction)

Some time back I made a notation I felt was capable of generating bewilderingly large ordinals, only to find out that it wasn't formal, and unfit to represent even $$\eta_0$$ ! (Or at least if it could, it required an amount of ingenuity I wasn't willing to expend at the time.)

Here is a revised and simplified version of my language.

Notation
$$'\text{ allows multiple variables to be names}$$

$$,\text{ separates two arguments or more in a function}$$

$$0\text{ is the base unit from which all other numbers originate, with the help of the following operator}$$

$$S(b)\text{ is the successorship of b, or b+1}$$

$$b\text{ is a variable which may only be a whole number or ordinal}$$

$$B\text{ is a function name}$$

$$\alpha+\beta\text{ is }\alpha\text{ added to }\beta$$

$$\alpha\cdot\beta\text{ is }\alpha\text{ multiplied by }\beta$$

$$\alpha\cdot\beta\text{ is }\alpha\text{ to the power of }\beta$$

$$:\text{ means 'such that'}$$

$$\text{ separates parts of a string from other parts to avoid ambiguity}$$

$$\{\phi\}_n\text{ is the n-th fixed point of the relation }\phi$$

$$\alpha=\beta\text{ implies that }\alpha\text{ and }\beta\text{ are equivalent }$$

$$\alpha>\beta\text{ implies that }\alpha\text{ is bigger than }\beta$$

$$\alpha<\beta\text{ implies that }\alpha\text{ is less than }\beta$$

\and\text{ is the logical operator 'and'}

\not\text{ is the logical complement 'not'}

$$\exists\text{ is the existential quantifier 'there exists'}$$

$$\forall\text{ is the universal quantifier 'for all'}$$

Example
Let us try to represent $$\omega$$ in the system. Here b is $$\omega$$.

$$\exists b:((b=\{B(b\}_1)\and(B(b')=S(0)+b'))$$

In words: "there exists a b such that b equals the first fixed point of the relation B(b), where B(b') equals 1 plus b'".

This was one example, but I'm sure you could go much farther with a sufficient number of characters. Revised LOOC is meant to be as basic as possible, not in the sense that it requires shorter strings to represent the same ordinals, but in the sense that at the basest level it is composed of easily understood fundamental operations which together can construct more complicated operations, which in turn form more complicated statements, and so forth.

There may be more refinements needed to make LOOC a completely formal language, but for now I am assured of it's ability to represent all ordinals definable using recursive formulas. If you have any recommendations or question please leave a comment below.