Editing Gödel's incompleteness theorems (section)

=== Relationship with the liar paradox ===

Gödel specifically cites [[Richard's paradox]] and the [[liar paradox]] as semantical analogues to his syntactical incompleteness result in the introductory section of "[[On Formally Undecidable Propositions in Principia Mathematica and Related Systems I]]".  The [[liar paradox]] is the sentence "This sentence is false."  An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence {{mvar|G}} for a system {{mvar|F}} makes a similar assertion to the liar sentence, but with truth replaced by provability: {{mvar|G}} says "{{mvar|G}} is not provable in the system {{mvar|F}}."  The analysis of the truth and provability of {{mvar|G}} is a formalized version of the analysis of the truth of the liar sentence.

It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "{{mvar|Q}} is the [[Gödel number]] of a false formula" cannot be represented as a formula of arithmetic. This result, known as [[Tarski's undefinability theorem]], was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, [[Alfred Tarski]].