On Sat, 30 Nov 2002, Peter Fairbrother wrote:
> Jim Choate wrote:
> >
> > With regard to completeness, I have Godel's paper ("On Formally
> > Undecidable Propositions of Principia Mathematica and Related Systems", K.
> > Godel, ISBN 0-486-66980-7 (Dover), $7 US) and if somebody happens to know
> > the section where he defines completeness I'll be happy to share it.
>
> That's* the wrong paper. You want "The completeness of the axioms of the
> functional calculus of logic" which is a 1930 rewrite of his doctoral
> dissertation. This is known as Godel's completeness theorem.
Actually it isn't since we -are- talking about incompleteness in the
larger discussion. In fact I took a few minutes to scan the paper
yesterday and found exactly what is needed at the bottem of the first
page, and top of the second page of the paper. In the Dover edition that's
pp 37/38 (or pp 173/174 in the original publication).
> Godel didn't invent the term though, and may not have said "this is the/my
> definition of completeness". I haven't read them for some time, and can't
> remember. He may well have assumed his readers would already know it.
Of course he didn't, he just made it irrelevant since you can't prove the
truthfullness of all the propositions requird to prove completeness.
Bottom line, mathematics may be complete but until somebody invents a
meta-mathematics broader than mathematics it will remain -an unprovable
proposition within mathematics, even in principle.-
Adios.
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