Editing Gödel's incompleteness theorems (section)

=== Construction of a statement about "provability" ===
Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this.

A formula {{math|''F''(''x'')}} that contains exactly one free variable {{mvar|x}} is called a ''statement form'' or ''class-sign''. As soon as {{mvar|x}} is replaced by a specific number, the statement form turns into a ''[[bona fide]]'' statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number {{mvar|n}}, {{tmath|F(n)}} is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as "2{{resx}}3 = 6".

Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form {{math|''F''(''x'')}} can be assigned a Gödel number denoted by {{math|'''G'''(''F'')}}. The choice of the free variable used in the form {{mvar|F}}({{mvar|x}}) is not relevant to the assignment of the Gödel number {{math|'''G'''(''F'')}}.

{{anchor|Bew}}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 {{mvar|p}}, one may ask whether a number {{mvar|x}} is the Gödel number of its proof. The relation between the Gödel number of {{mvar|p}} and {{mvar|x}}, the potential Gödel number of its proof, is an arithmetical relation between two numbers.  Therefore, there is a statement form {{math|''Bew''(''y'')}} that uses this arithmetical relation to state that a Gödel number of a proof of {{mvar|y}} exists:
:{{math|1=''Bew''(''y'') = ∃ ''x''}} ({{mvar|y}} is the Gödel number of a formula and {{mvar|x}} is the Gödel number of a proof of the formula encoded by {{mvar|y}}).

The name '''Bew''' is short for ''beweisbar'', the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described.  Note that "{{math|''Bew''(''y'')}}" is merely an abbreviation that represents a particular, very long, formula in the original language of {{mvar|T}}; the string "{{math|Bew}}" itself is not claimed to be part of this language.

An important feature of the formula {{math|''Bew''(''y'')}} is that if a statement {{mvar|p}} is provable in the system then {{math|''Bew''('''G'''(''p''))}} is also provable. This is because any proof of {{mvar|p}} would have a corresponding Gödel number, the existence of which causes {{math|Bew('''G'''(''p''))}} to be satisfied.