What is Gödel's incompleteness theorems?
'''Gödel's incompleteness theorems''' is the name given to two [[theorem]]s (true mathematical statements), proved by [[Kurt Gödel]] in 1931. They are theorems in [[mathematical logic]]. Mathematicians once thought that everything that is true has a [[mathematical proof]]. A system that has this property is called ''complete''; one that does not is called ''incomplete''. Also, mathematical ideas should not have ''contradictions''. This means that they should not be true and false at the same time. A system that does not include contradictions is called ''consistent''. A system is a collection of theorems (logical consequences) based on [[Axiom|axioms]] (basic assumptions). Axioms are statements that are accepted as true, and need no proof. Gödel said that ''every non-trivial formal system (consistent and axiomatic system with theorems listable by following an algorithm) is incomplete and not provably consistent'':<ref>Nagel, Ernest; Newman, James Roy & Hofstadter, Douglas [1958] 2002. ''Gödel's proof'', revised ed. {{ISBN|0-8147-5816-9}} [http://www.ams.org/mathscinet-getitem?mr=2002i:03001] </ref><ref>{{Cite web|date=2023-09-07|title=Quanta Magazine|url=https://www.quantamagazine.org/how-godels-proof-works-20200714/|access-date=2023-09-23|website=web.archive.org|archive-date=2023-09-07|archive-url=https://web.archive.org/web/20230907041330/https://www.quantamagazine.org/how-godels-proof-works-20200714/|url-status=bot: unknown}}</ref> #There will always be questions that cannot be answered, using a certain set of axioms; there are truths that cannot be proved using the axioms of the system. #You cannot prove that a system of axioms is consistent according to the axioms of the system. '''Outline of Proof of Gödel’s Theorems''' The proof of Gödel’s theorems is based on forming and considering [https://web.archive.org/web/20230907041330/https://www.quantamagazine.org/how-godels-proof-works-20200714/ special self-referential statements]. One way to prove the theorems is to consider the first theorem before cons