Editing Liar paradox (section)

==Applications==

===Gödel's first incompleteness theorem===

[[Gödel's incompleteness theorems]] are two fundamental theorems of [[mathematical logic]] which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by [[Kurt Gödel]] in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the [[first incompleteness theorem]], Gödel used a modified version of the liar paradox, replacing "this sentence is false"  with "this sentence is not provable", called the "Gödel sentence G". His proof showed that for any sufficiently powerful theory T, G is true, but not provable in T. The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.<ref>{{cite book | last1=Crossley | first1=J.N. | last2=Ash | first2=C.J. | last3=Brickhill | first3=C.J. | last4=Stillwell | first4=J.C. | last5=Williams | first5=N.H. | title=What is mathematical logic? | zbl=0251.02001 | location=London-Oxford-New York | publisher=[[Oxford University Press]] | year=1972 | isbn=978-0-19-888087-5 | pages=52–53 }}</ref>

To prove the first incompleteness theorem, Gödel represented [[Gödel numbering|statements by numbers]]. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number.

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) and by [[Alfred Tarski]].

[[George Boolos]] has since sketched an alternative proof of the first incompleteness theorem that uses [[Berry's paradox]] rather than the liar paradox to construct a true but unprovable formula.