Gödel’s First Incompleteness Theorem in Relation to the Liar Paradox

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.”

Kurt Gödel

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.

3 thoughts on “Gödel’s First Incompleteness Theorem in Relation to the Liar Paradox

  1. Pingback: Atheism Violates The Laws Of Reason And Logic

  2. In my interpretation of truth, provability, what constitutes each, so forth, the debate on the liars paradox is a moot one.

    Does a question have a truth value? Does an exclamation? The point is, in language, some sentences, though consistence and which make perfect sense, wont necessarily have truth values to them. It stands to reason then that there may indeed exist some statements of the sort.

    A statement, in language, is one that provides usable, consistent, and meaningful information. The liars sentence does none of that. It is a sentence for the sake of sentences. Its no more meaningful than had you tossed words together out of the dictionary quite randomly. In fact, arguably, there is more meaning in simply remaining silent.

    The liars paradox has no resolution, because there is nothing to resolve. Its nonsense from ground up. There is no dilemma and there is no inconsistency. Its a hypothetical situation that cannot and does not exist. This is like debating over how much paint it will take to paint Gabriel’s Horn. If something cannot exist in reality then debating its traits, or any practical use made of it, is nonsense.

    The paradox is a phenomenon of language, a hypothetical situation invented solely to confuse, because it is a contradiction. And we all know that contradictions cannot exist in reality, so searching for a resolution is moot, pointless.

    There is a difference between a hypothetical contradiction and an actual contradiction. In logic, it is not unusual to assume a faulty premise, arrive at a contradiction, and then reject that faulty premise as a result. Proof by contradiction is a valid tool because, in the end, you are left with one consistent and true statement. The Liars Paradox, however, asserts that the contradiction exists in actuality.

    Those are my thoughts.

  3. Pingback: Kurt Gödel « Earthpages.ca

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