Liar paradox

In philosophy and logic, the liar paradox encompasses paradoxical statements such as:

Gödel's theorem

The proof of Gödel's incompleteness theorem uses self-referential statements that are similar to the statements at work in the Liar paradox.

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In the context of a sufficiently strong axiomatic system A of arithmetic:

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:(1) This statement is not provable in A.

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You will notice that (1) does not mention truth at all (only provability) but the parallel is clear. Suppose (1) is provable, then what it says of itself, that it is not provable, is true. But this conclusion is contrary to our supposition, so our supposition that (1) is provable must be false. Suppose the contrary that (1) is not provable, then what it says of itself is true, although we cannot prove it. Therefore, there is no proof that (1) is provable, and there is also no proof that its negation is provable (i.e., there is no proof that it is also unprovable). Whence, A is incomplete because it cannot prove all truths, namely, (1) and its negation. Statements like (1) are called undecidable. We take for granted that all provable statements are true, but Gödel showed that the converse, that all true statements are provable in some one system is not the case. (This does not mean that all true statements are not provable in some system or other.)

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Tarski's theorem, closely related to Gödel's Theorem, is a more direct application of the Liar Paradox, though there is no actual paradox involved; instead, the "paradox" simply demonstrates that all the true sentences of arithmetic are not arithmetically definable (or that arithmetic cannot define its own truth predicate; or that arithmetic is not "semantically closed").

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