 One version of Gödel’s first incompleteness theorem states that for any consistent axiomatic theory of arithmetic, which can be recognised to be sound, there will be an arithmetic truth—viz., its Gödel sentence—not provable in it, but which can be established as true by intuitively correct reasoning. The heart of Gödel’s theorem is, in fact, a paradox that concerns the sentence, \(G\), ‘This sentence is not provable’. If \(G\) is provable, then it is true and so not provable. Thus \(G\) is proved. Hence \(G\) is true and so unprovable. If an underlying paraconsistent logic is used to formalise the arithmetic, and the theory therefore allowed to be inconsistent, the Gödel sentence may well be provable in the theory (essentially by the above reasoning). So a paraconsistent approach to arithmetic overcomes the limitations of arithmetic that are supposed (by many) to follow from Gödel’s theorem. For other ‘limitative’ theorems of metamathematics, see Priest 2002. For some of the original work by Meyer applying paraconsistent logic to arithmetic, and new commentaries and developments, see the collection Ferguson and Priest 2021.