Gödel's incompleteness theorems

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Gödel's incompleteness theorems are twa theorems o mathematical logic that demonstrate the inherent leemitations o ivery formal axiomatic seestem capable o modellin basic arithmetic.

The first incompleteness theorem states that na consistent seestem o axioms that's theorems can be leetit bi an effective procedure (i.e., an algorithm) is capable o pruivin aw truths aboot the arithmetic o the naitural nummers. For ony sic consistent formal seestem, thare will ayeweys be statements aboot the naitural nummers that are true, but that are unpruivable within the seestem. The seicont incompleteness theorem, an extension o the first, shaws that the seestem canna demonstrate its awn consistency.