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Wikipedia: Liar paradox
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The liar paradox, attributed to the Greek philosopher [Eubulides of Miletus]? who lived in the fourth century BC, is the statement
- I am lying now.
or more succinctly
- This statement is false.
As opposed to the Epimenides paradox, this is a true paradox: assuming that the statement is true, then it must be false; assuming it is false, then it must be true. No truth value can be consistently assigned to the statement.
The proof of Goedels Incompleteness Theorem essentially consists in a
formally correct formulation of a variation of this paradox in the context of a sufficiently strong [axiomatic system]?.
To avoid having a sentence refer to its own truth value,
one can also construct the paradox
- The following sentence is true.
- The preceding sentense is false.