User:Physis/Canonical calculus

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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.

Contents

[edit] Formal definition

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

[edit] Syntax

<calculus> ::= rule
<calculus> ::= <calculus> <rule>

<rule> ::= <term>
<rule> ::= term \to <rule>

<term> ::= <symb>*

<symbol> ::= <variable>
<symbol> ::= <nonlog>

[edit] 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

[edit] References

  • Ruzsa, Imre (1988). Logikai szintaxis és szemantika I. Budapest: Akadémiai Kiadó.