[FOM] On the Scope of Predicative Reasoning

Dmytro Taranovsky dmytro at MIT.EDU
Tue Apr 11 14:43:19 EDT 2006

In the debate between Harvey Friedman and Nik Weaver, there are really
two different questions:
1.  Which formal system best corresponds to predicativity?
2.  Which arithmetical statements are predicatively demonstrable?

In (2), predicativity refers to the philosophical stance rather than a
formal system.  

I think that no recursive ordinal above omega is fundamentally less
predicative than every preceding ordinal and that no recursive ordinal
is 100% impredicative.  Because (appearance of) impredicativity arises
gradually, a formal system should not be required to encompass all
predicative reasoning but just be a good fit, taking mathematical
practice into account.  In this sense, Feferman/Schutte analysis may
have the best correspondence with predicativity.

The second question splits according the nature of the demonstration
required.  If we allow heuristic analysis, then consistency of ZFC can
be demonstrated finitistically.  Instead, what we want is a proof,
specifically a demonstration that has a certain type of certainty (and
would ideally be infallible).  We do not know exactly what that means,
so we cannot rule out that Con(ZFC) is finitistically provable.  

However, we can look at proposed proofs on a case-by-case basis and see
whether they are proofs.  Nik Weaver argues that the philosophical
position of predicativism entails the Kruskal Tree Theorem.  That
argument should not be dismissed just because Feferman/Schutte analysis
may have the best fit with predicativity. Note, however, that the
philosophical position of predicativism is vague, so such an argument
should either clarify the position or succeed from the common ground of
the philosophical variations of predicativity.

Dmytro Taranovsky

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