Gödel's first incompleteness theorem states that all consistent axiomatic formulations of number theory which include
Peano arithmetic include undecidable propositions (Hofstadter 1989). This 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).
The inclusion of Peano arithmetic is needed, since for example Presburger arithmetic is a consistent axiomatic formulation of number theory, but it is decidable.
However, Gödel's first incompleteness theorem also holds for Robinson arithmetic (though Robinson's result came much later and was proved by Robinson).
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.
