On the circularity of Gödel's incompleteness proofs

There is a consensus that Gödel’s incompleteness proofs are some of the most important theorems of mathematical logic. This paper questions this general assumption by arguing that incompleteness proofs are antinomial. The main criticism is that Gödel's reasoning is viciously circular since it relies on diagonalization, which is an indirect self-reference procedure that should be disallowed by a hierarchy of formal systems. It is a mistake to allow a self-referential formula such as S ≡ ∼Bew [R(n);n] within the system, since provability is a meta-predicate that requires a system of higher order. The requirements used in semantics to prevent semantic paradoxes can be adapted to syntactics to prevent syntactical paradoxes, such as the Gödel sentence that says of itself that it is unprovable. This undermines Gödel's proofs.