Liar paradox

The liar paradox, attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century B.C., is the paradoxical statement

I am lying now.

or

This statement is false.

As opposed to the Epimenides paradox, this statement is indeed paradoxical: assuming that the statement is true, then it must be false; assuming it is false, then it is not false. No truth value can be consistently assigned to the statement.

Even the conclusion that the statement is neither true nor false leads to a contradiction: the statement claims to be false, but isn't, so it claims a falsehood and is therefore false.

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.

The proof of Gödel's incompleteness theorem essentially consists of a formally correct formulation of a variation of this paradox in the context of a sufficiently strong axiomatic system A:

A proof exists in A that this sentence is false.

If a proof exists using only the axioms in A that the statement is true, then this implies that there is also a proof that the statement is false. Conversely, if a proof exists in A that the statement is false, then this proof is an example showing that the statement is true. Thus, if a proof exists either way, the system is inconsistent, in that a single statement can be proven to be both true and false.

On the other hand, if there exists no proof in A of the statement either way, then no contradiction arises. The system A is called incomplete in this case: there exists a statement which can neither be proven nor disproven in A.

Similarly, by using the statement "No proof exists in A that this statement is true", we can see that in a consistent system there are statements that are "clearly" true, which cannot be proven to be so in A.