Gödel’s first incompleteness theorem states that: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”
How does Gödel’s first incompleteness theorem relate to the liar paradox?
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 G for a theory T makes a similar assertion to the liar sentence, but with truth replaced by provability: G says “G is not provable in the theory T.” The analysis of the truth and provability of 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 “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 by Gödel when he was working on the proof of the incompleteness theorem.