Editing Gödel's incompleteness theorems (section)

=== Truth of the Gödel sentence ===

The first incompleteness theorem shows that the Gödel sentence {{math|''G''<sub>''F''</sub>}} of an appropriate formal theory {{mvar|F}} is unprovable in {{mvar|F}}. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true ({{harvnb|Smoryński|1977|p=825}}; also see {{harvnb|Franzén|2005|pp=28–33}}). For this reason, the sentence {{math|''G''<sub>''F''</sub>}} is often said to be "true but unprovable." {{harv|Raatikainen|2020}}.  However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence {{math|''G''<sub>''F''</sub>}} may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as [[primitive recursive arithmetic]], which proves the implication {{math|''Con''(''F'')→''G''<sub>F</sub>}}, where {{math|''Con''(''F'')}} is a canonical sentence asserting the consistency of {{mvar|F}} ({{harvnb|Smoryński|1977|p=840}}, {{harvnb|Kikuchi|Tanaka|1994|p=403}}).

Although the Gödel sentence of a consistent theory is true as a statement about the [[intended interpretation]] of arithmetic, the Gödel sentence will be false in some [[Peano axioms#Nonstandard models|nonstandard models of arithmetic]], as a consequence of Gödel's [[completeness theorem]] {{harv|Franzén|2005|p=135}}.  That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false.  As described earlier, the Gödel sentence of a system {{mvar|F}} is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system {{mvar|F}}, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" &ndash; it must contain elements that do not correspond to any standard natural number ({{harvnb|Raatikainen|2020}}, {{harvnb|Franzén|2005|p=135}}).