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

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

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

<term> ::= <symb>*

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

Motivating example

Decimal form of 3-divisible natural numbers

$x\to x0$

$x\to x3$

$x\to x6$

$x\to x9$

$x\to x\mathbf {U} y\to y2$

$0\mathbf {U} 1$

$1\mathbf {U} 2$

$2\mathbf {U} 3$

$3\mathbf {U} 4$

$4\mathbf {U} 5$

$5\mathbf {U} 6$

$6\mathbf {U} 7$

$7\mathbf {U} 8$

$8\mathbf {U} 9$

$9\mathbf {U}$

References

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