Editing Gödel's incompleteness theorems (section)

=== Syntactic form of the Gödel sentence ===

The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in {{mvar|F}}. However, the sequence of steps is such that the constructed sentence turns out to be {{math|''G''<sub>''F''</sub>}} itself. In this way, the Gödel sentence {{math|''G''<sub>''F''</sub>}} indirectly states its own unprovability within {{mvar|F}}.<ref>{{harvnb|Smith|2007|p=135}}.</ref>

To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on [[Gödel number]]s of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.

Thus, although the Gödel sentence refers indirectly to sentences of the system {{mvar|F}}, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a [[primitive recursive function|primitive recursive]] relation {{harv|Smith|2007|p=141}}.   As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at level <math>\Pi^0_1</math> of the [[arithmetical hierarchy]]).  Via the [[MRDP theorem]], the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables {{harv|Franzén|2005|p=71}}.