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...n.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems incompleteness theorems], which showed that sufficiently strong [[proof]] theories cannot prove the ...ong effective theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and [https://en.wi

:* '''[[Consistency proof|Consistency]]''', which means that none of the theorems of the system contradict one another. ...rom a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.