Gödel numbering
A Gödel numbering is any way of assigning numbers to the formulas^{} of a language^{}. This is often useful in allowing sentences^{} of a language to be selfreferential. The number associated with a formula $\varphi $ is called its Gödel number and is denoted $\mathrm{\u231c}\varphi \mathrm{\u231d}$.
More formally, if $\mathcal{L}$ is a language and $\mathcal{G}$ is a surjective^{} partial function^{} from the terms of $\mathcal{L}$ to the formulas over $\mathcal{L}$ then $\mathcal{G}$ is a Gödel numbering. $\mathrm{\u231c}\varphi \mathrm{\u231d}$ may be any term $t$ such that $\mathcal{G}(t)=\varphi $. Note that $\mathcal{G}$ is not defined within $\mathcal{L}$ (there is no formula or object of $\mathcal{L}$ representing $\mathcal{G}$), however properties of it (such as being in the domain of $\mathcal{G}$, being a subformula, and so on) are.
Athough anything meeting the properties above is a Gödel numbering, depending on the specific language and usage, any of the following properties may also be desired (and can often be found if more effort is put into the numbering):

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If $\varphi $ is a subformula of $\psi $ then $$

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For every number $n$, there is some $\varphi $ such that $\mathrm{\u231c}\varphi \mathrm{\u231d}=n$

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$\mathcal{G}$ is injective^{}
Title  Gödel numbering 

Canonical name  GodelNumbering 
Date of creation  20130322 12:58:21 
Last modified on  20130322 12:58:21 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  8 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 03B10 
Related topic  BeyondFormalism 
Defines  Gödel number 