Comment by swatow
11 years ago
Goedel's incompleteness theorem uses Peano Arithmetic, which Nelson claims is (possibly) inconsistent. He develops his own "predicative" arithmetic, to which Goedel's theorem does not apply.
(large pdf) https://web.math.princeton.edu/~nelson/books/pa.pdf
I'll read it, thanks. I always (layman) understood Goedel's theorem as meaning you can be complete or consistent but not both. So you can have a language that can describe everything but you'll have paradoxes, or you'll have no paradoxes but not be able to describe everything (all possibilities) in your system
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.
Wow. I have a lot of reading to do. Thanks!