[FoRK] Gödel's theorem (was: Anything to be learned from religion?)
Mon Aug 15 16:08:13 PDT 2005
>What I'm getting at is if mathemeticians can (historically, that is -- I'm
>not sure if they still are) be deeply and profoundly disturbed at the mere
>thought that no formal system can contain all true statements about itself
>(at least, I think that's what Godel implies...
I'm not sure what you mean by "contained in." Gödel's
theorem about the incompleteness of arithmetic says
any finite formalization of arithmetic fails to PROVE
some of the truths of arithmetic. Second-order
arithmetic "contains" all the truths, in the sense that
it has only one model, up to isomorphism.
>.. then doesn't it lend weight to metaphysics ..
Not that I can see.
>.. to suppose that there is truth (little t) outside of
Now, what does that have to do with metaphysics?
There are many things we don't know, and that we
won't ever know. Gödel's theorem surprised only in
proving that some of mathematics also includes
things that never will be known. (a) If you think
about it, the fact of that may be less surprising than
that it can be rigorously proven. (b) Not everything
is as deep as numbers. There are many kinds of
mathematics that are complete. Euclidean geometry.
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