[FOM] Martin-Lof predicativity
William Tait
wwtx at earthlink.net
Tue Apr 11 23:25:50 EDT 2006
In view of my previous note today (April 11) on Martin-Loef's type
theory and, in particular, my statement that it is impredicative,
some further comment is needed in responsde to the quote below from
Jesper Carlstroem:
Predicativity is a property of definitions. Impredicative
definitions of functions are admitted in the M-L theory, as I noted.
But all of the functions of finite type definable by impredicative
primitive recursion---indeed, all of the objects definable in the
Curry-Howard theory---are also definable by predicative recursion on
epsilon_0 (in the F_S sense). I did not know that Feferman had shown
that all of the objects defined in M-L's theory, e.g. those defined
using reflection, also have predicative definitions, although it is
not surprising. {Perhaps it was forshadowed by Matin-Loef's statement
that the stregnth of his theory should be close to that of
predicative analysis.
Bill Tait
On Apr 11, 2006, at 2:14 PM, Jesper Carlström wrote:
> I don't have the paper at hands, but I got a useful comment from
> Thierry
> Coquand. He says that the type theory presented in this paper
> [Martin-Loef's] is, in
> fact, predicative in the sense of Feferman/Schutte. However, it was
> not
> known to be by that time (although conjectured by Peter Hancock), but
> proved later by Feferman.
> http://www.cs.nyu.edu/mailman/listinfo/fom
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