Comment by swatow

Yes, Goedel's Theorem says you can be complete or consistent but not both, but only about systems that are stronger than Peano Arithmetic, i.e. systems that contain the axioms of Peano Arithmetic, as well as any other axioms.

Usually this is left out, because Peano Arithmetic is treated as a MVP for mathematics. But Nelson claims Peano Arithmetic may be inconsistent, and proposes a weaker system.