FOM: applications of proof theory
Stephen G Simpson
simpson at math.psu.edu
Thu Apr 9 13:57:30 EDT 1998
Bill Tait 9 Apr 98 11:11:03:
> >to find a constructive interpretation (in some loose sense, an extension
> >of finitistic mathematics) of parts of classical mathematics.
OK. But then, do you view such an interpretation as a consistency
proof? My point is that it seems better to view it as a
proof-theoretic reduction and to formulate the result as a
conservation theorem. Do you agree? Or, how would you formulate it?
For example, you said:
> >So for me, it wasn't the well-orderedness of the ordering of the natural
> >numbers which carried constructive conviction; it was the constructive
> >theory of iterated inductive definition and, in particular, the
> >constructive higher number classes.
and I would like it better if you had stated a conservativity result,
along the lines of "classical ID_{<omega} is conservative over
intuitionistic ID_{<omega} for a certain class of sentences." (I'm
not sure exactly what the right class of sentences is.) Don't you
agree that this kind of formulation is more informative?
I guess my real point is that proof theory qua search for finitistic
consistency proofs a la Hilbert is dead and buried, as a result of
G"odel's 2nd incompleteness theorem. So we need to look elsewhere for
foundationally significant applications of technical work in proof
theory. My favorite places to look are: (1) combinatorial
independence results, (2) reductions formulated as conservation
theorems. What are your favorite places to look?
-- Steve
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