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- W2009171410 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 Π 1 0 -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 Π 1 1 -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 Π 1 0 formula A which defines a nonrecursive set, since then some instance A [ x / n ] must be undecided and A [ x / n ] is a closed Π 1 0 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 Π 1 1 -paths. A partially reckonable function always has a Π 1 1 graph (and if its graph is Δ 1 1 then it is by C. Jockusch, r.e.; so some Π 1 1 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." @default.
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- W2009171410 title "Which number theoretic problems can be solved in recursive progressions on Π<sub>1</sub><sup>1</sup>-paths through <i>O</i>?" @default.
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- W2009171410 doi "https://doi.org/10.2307/2272975" @default.
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