Which number theoretic problems can be solved in recursive progressions on Π11-paths through O?

Abstract

This note is an addendum to an incompleteness result of Feferman and Spector [1]: for any such progression satisfying the (natural) conditions listed in [1] there is a true Π10-formula which cannot be derived in the progression. The idea of their proof yields a stronger result.If each numerical instance A[x/n] of the formula A (with the free numerical variable x) is decided in a recursive progression along a Π11-path then the set {n: ⊦ A[x/n]} is recursive.In other words, only recursively solvable problems are decidable in such progressions. This yields the incompleteness result of [1] when specialized to some Π10 formula A which defines a nonrecursive set, since then some instance A[x/n] must be undecided and A[x/n] is a closed Π10 formula. The stronger result is needed to extend Church's thesis for total functions reckonable in formal systems to (total) functions reckonable in recursive progressions on Π11-paths. A partially reckonable function always has a Π11 graph (and if its graph is Δ11 then it is by C. Jockusch, r.e.; so some Π11 sets are not partially reckonable). The relevance of these facts for the model of mathematical reasoning provided by (variants of) Turing's ordinal logics is analyzed in (b) (ii) of Part II, which goes into our current knowledge about Church's thesis.