Informally, Gödel's incompleteness theorem states that all consistent axiomatic formulations of number theory include undecidable propositions (Hofstadter 1989). This is sometimes called Gödel's first incompleteness theorem, and answers in the negative Hilbert's problem asking whether mathematics is "complete" (in the sense that every statement in the language of number theory can be either proved or disproved).
A statement sometimes known as Gödel's second incompleteness theorem states that if number theory is consistent, then a proof of this fact does not exist using the methods of first-order predicate calculus. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.
Gerhard Gentzen showed that the consistency and completeness of arithmetic can be proved if transfinite induction is used. However, this approach does not allow proof of the consistency of all mathematics. --http://mathworld.wolfram.com/
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