[FOM] on Bas Spitters on "constructive impredicativity?"
friedman at math.ohio-state.edu
Thu Mar 30 19:25:06 EST 2006
On 3/30/06 1:26 PM, "Bas Spitters" <spitters at cs.ru.nl> wrote:
> There is now also an intuitionistic version, i.e. provable in Kleene-Vesley's
> FIM, by Veldman.
> (or Wim Veldman: An intuitionistic proof of Kruskal's theorem. Arch. Math.
> Log. 43(2): 215-264 (2004))
> This result was elicited by Coquand's result. Veldman's proof is predicative
> and can be translated into a Bishop-style constructive proof. There are many
> nice constructive results in this area (like the ones connected to open
> induction) that I do not want to go into now.
There is no predicative proof of Kruskal's theorem, under the classical
Feferman/Schutte elucidation. So if you say that Veldman's proof is
predicative, then you must be referring to some nonstandard analysis of
predicativity. Which nonstandard form?
I always viewed Bishop style constructivity as entirely predicative under
the Feferman/Schutte elucidation. In fact, I always viewed Bishop style
constructivity as living comfortably within a conservative extension of HA,
and so once again Kruskal's theorem cannot be proved there either. Where in
Bishop do you see any substantial logical strength beyond that of HA?
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