Liar paradox

The liar paradox, attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century B.C., 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 Gödels Incompleteness Theorem essentially consists of 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 sentence is false.