[FOM] Hilbert's proof
Alasdair Urquhart
urquhart at cs.toronto.edu
Tue Mar 9 11:34:17 EST 2010
> > What was an essence of the "totally new method" by which Hilbert could
> > prove his the existence of a finite basis for the invariant forms?
> >
> > with reagards, adam
>
> It was non-constructive. There is a reasonable brief discussion of this on
> Wikipedia: http://en.wikipedia.org/wiki/David_Hilbert#The_finiteness_theorem
I think it depends on what you mean by "non-constructive". Hilbert's
second proof is, I believe, formalizable in the theory RCA_0 +
"omega^omega is well-ordered." It is not provable in RCA_0.
For these and other results see Steve Simpson's fine article
"Ordinal Numbers and the Hilbert Basis Theorem", JSL Volume 53 (1988),
pp. 961-974.
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