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

  • G. Kreisel (a1)

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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.

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[1]Feferman, S. and Spector, C., Incompleteness along paths in progressions of theories, this Journal, vol. 27 (1962), pp. 383390.
[2]Friedbero, R. M. and Rogers, H., Reducibility and completeness of sets of integers, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 117125.
[3]Gandy, R. O., The concept of computability, Scientific thought 1900–1960, Clarendon Press, Oxford, 1969, pp. 14.
[4]Kreisel, G., Mathematical logic: what has it done for the philosophy of mathematics, Bertrand Russell, philosopher of the century, Academic Press, London, 1967, pp. 201272, 315–316.
[5]Kreisel, G., Hubert's programme and the search for automatic proof procedures, Springer Lecture Notes, vol. 125 (1970), pp. 128146; reviewed in Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 206 (1971), pp. 277–278.
[6]Kreisel, G., A Church's thesis: A kind of reducibility axiom for constructive mathematics, Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 121150; reviewed in Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 199 (1971), pp. 300–301.
[7]Kreisel, G., Some reasons for generalizing recursion theory, Logic Colloquium '69, North-Holland, Amsterdam, 1971, pp. 139198; reviewed in Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 219 (1972), pp. 17–19.

{Material in the dissertation of S. G. Simpson, M.I.T., 1971, supersedes p. 159, 1.–16; p. 161, 11. 5–10; p. 172, 11. 11–20. Material in [11] supersedes the following passages which are directly relevant to the present note. Ad p. 177, 1.–9 on Church's superthesis, see p. viii of the dissertation, ad p. 187 concerning total versus partial functions, see p. 6. Also the suggestion on p. 193, 1. 20 is realized by the normed uniformly reflexive structures on p. 4, loc. cit.}

[8]Kreisel, G., A survey of proof theory. II, Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 109170. Corrections. P. 147, 1. 14, replace by the stronger principle

cf. §4 of the present paper. P. 164, 1. 11 replace ‘view’ by ‘review’; p. 169, 1. 15 add ‘reviewed in this Journal, vol. 35 (1970), 330–33’; 1. 8 replace ‘to appear’ by ‘238–265’.

[9]Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, vol. 42 (1936/1937), pp. 230265.
[10]Feferman, S., Transfinite recursive progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259316.
[11]Barendregt, H. P., Supplementary part II of dissertation, Utrecht, 1971.
[12]Troelstra, A. S., Principles of intuitionism, Springer Lecture Notes, vol. 95 (1969).
[13]Troelstra, A. S., Notions of realizability for intuitionistic arithmetic and intuitionistic arithmetic in all finite types, Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 369405.

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

  • G. Kreisel (a1)

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