#MathsInMinutes Day 31: Gödel´s incompleteness theorems
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Gödel´s incompleteness theorems are two remarkable results that changed how mathematicians view axiomatic mathematics. Developed by German mathematician Kurt Gödel in the late 1920s and early 1930s, the theorems grew out of his method for coding statements in axiomatic theories, and for showing how statements could be modified by logical rules.
Although the axiomatic method for describing various fields of mathematics had proven highly successful, some theories had been shown to require infinite sets of axioms in themselves, and therefore mathematicians were anxious to find formal ways of proving the completeness and consistency of a given set of axioms.
A set of axioms is held to be complete if it is capable of proving or negating any statement given in its appropriate language, while a set of axioms is consistent if no statement can be made that can be both proved and negated.
Gödel´s first theorem states that:
In any (appropriate) axiomatic theory, there exists statements which make sense within the theory but which cannot be proved true or false within that theory.
This means that the axioms of a theory, which we might hope to describe that theory completely, can never do this and that it is always possible to augment the number of axioms.
As if this wasn't bad enough, a second complication involved the internal consistency of sets of axioms:
It is only possible to prove that an (appropriate) set of axioms is inconsistent, and not that they are consistent.
In other words, we can never be sure that a set of axioms does not contain a hidden contradiction.
Gödel´s results have profound implications for the philosophy of mathematics – but, in general, working mathematicians tended to carry on as a thought nothing had changed.
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