User:Physis/Canonical calculus

Canonical calculus is a formal way to derive the words of a formal language. It is motivated by the pattern of inductive definitions.

It can be used as one tool in the approach of building logic without circularity: "taming it to spiral".

Hypercalculus is a special instance of it, which enables an especially concise approach to self-referent theorems, and formulation of Gödel's incompleteness theorem.

Formal definition

 * $$\left\langle\mathrm{Var}, \mathrm{NonLog}\right\rangle$$

Syntax
::=







Motivating example
Decimal form of 3-divisible natural numbers


 * $$x \to x0$$
 * $$x \to x3$$
 * $$x \to x6$$
 * $$x \to x9$$
 * $$x \to x \mathbf U y \to y2$$


 * $$0 \mathbf U 1$$
 * $$1 \mathbf U 2$$
 * $$2 \mathbf U 3$$
 * $$3 \mathbf U 4$$
 * $$4 \mathbf U 5$$
 * $$5 \mathbf U 6$$
 * $$6 \mathbf U 7$$
 * $$7 \mathbf U 8$$
 * $$8 \mathbf U 9$$
 * $$9 \mathbf U $$