The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement , one may ask whether a number is the Gödel number of its proof. The relation between the Gödel number of and , the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form that uses this arithmetical relation to state that a Gödel number of a proof of exists: ( is the Gödel number of a formula and is the Gödel number of a proof of the formula encoded by ).
The notion of provability itself can also be encoded by Gödel numbers in the following way since a proof is a list of statements which obey certain rules the Gödel number of a proof can be defined Now for every statement  one may ask whether a number is the Gödel number of its proof The relation between the Gödel number of and  the potential Gödel number of its proof is an arithmetical relation between two numbers Therefore there is a statement form that uses this arithmetical relation to state that a Gödel number of a proof of exists ( is the Gödel number of a formula and is the Gödel number of a proof of the formula encoded by )