Comment by hellbanner

11 years ago

I'll read the paper, it sounds interersting. But isn't all Math "circular" -- Goedels Second Incompleteness Theorem yeah?

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_the...

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hellbanner

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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

  • hellbanner  11 years ago

    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

    • swatow  11 years ago

      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.

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lmm  11 years ago

Even if an axiom system could prove its own consistency, that wouldn't be any less circular - we could believe T because T proves that T is consistent - but if T were inconsistent then it might still prove that T was consistent.

Most of us take some axioms and believe them, ultimately because they correspond to physical experience - if you take a pile of n pebbles and add another pebble to the pile, you get a pile of n+1 pebbles, and n+1 is always a new number that doesn't =0 (that is, our pile looks different from a pile of 0 pebbles). Maybe there's some (very large) special n where this wouldn't happen - where you'd add 1 and get a pile of 0 pebbles, or where you can have n pebbles but you can't have n sheep. But so far we haven't found that. So far the universe remains simple, and the axioms of arithmetic are simpler than conceivable alternatives.