Skip to main content
SpringerLink
Log in

Hyperarithmetical Sets

  • Chapter
  • First Online:
Book cover Martin Davis on Computability, Computational Logic, and Mathematical Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 10))

Abstract

The hyperarithmetical sets of natural numbers were introduced (independently) in the early 1950s by Martin Davis, Andrej Mostowski and Stephen Cole Kleene and their study is surely one of the most significant developments in the theory of computability: they have a rich and interesting structure and they have found applications to many areas of mathematics, including inductive definability , higher-type recursion, descriptive set theory and even classical analysis. This article surveys the development of the subject in its formative period from 1950 to 1960, starting with a discussion of its origins and with some brief pointers to later developments. There are few proofs, chosen partly because of the importance of the results but mostly because they illustrate simple, classical methods specific to this area which are not easy to find in the literature, especially in the treatment of uniformity; and these are given in the spirit (if not the letter) of the methods which were available at the time. This is an elementary, expository article and includes an Appendix which summarizes the few basic facts about computability theory that it assumes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    cf.  Moschovakis [32].

  2. 2.

    Kleene had presented much of his [15] in a meeting of the American Mathematical Society in September 1940. I do not know when Post obtained the results in his [42].

  3. 3.

    For example, Davis [4] proves that the set \(\{e :(\forall x)[\{e\}(x)\downarrow ]\}\) of codes of total recursive functions is in \(\varPi ^0_2\setminus \Sigma ^0_2\). The Hierarchy Theorem also yields a trivial proof of Tarski’s Theorem for \(\mathbf{N}\), that arithmetical truth is not arithmetical .

  4. 4.

    He also said that “...with a few exceptions explicitly so noted, we have obtained formal proofs of all the consequently mathematical theorems here developed informally”, and it is clear that the purely intuitive approach can only go so far: we cannot hope to prove that (say) the word problem for semigroups is unsolvable on the basis of our intuitions about computability, without a rigorous definition of recursive functions and an appeal to the Church-Turing Thesis.

  5. 5.

    For completeness, we will repeat in this section some parts of §7–§9 of Moschovakis [37], which goes over some of the same ground in more detail and includes several proofs.

  6. 6.

    For example, to prove that \(K_k\) is \(\Sigma ^0_k\)-complete, you need the first of the following strengthenings of  (5.10): there are recursive injections u(e, t), v(e) such that for all A, B and all e, t,

    $$\begin{aligned} (1)~~\{e\}^A(t)\downarrow \iff u(e,t)\in A' \text { and }(2)~~A\leqslant ^T_e B \implies A'\leqslant ^1_{v(e)}B'. \end{aligned}$$
    (5.15)

    Proof: For (1), choose \(\overline{m}\) so that for any A, \(\{\overline{m}\}^A(e,t,y) =\{e\}^A(t)\) and set \(u(e,t) = S^2_1(\overline{m},e,t)\). For (2) you start with a recursive \(v_1(e)\) such that \(A\leqslant ^T_e B \implies \{e\}^A(t) = \{v_1(e)\}^B(t)\) and do a similar construction. That u(e, t) and v(e) are (absolutely) recursive injections—which has applications—depends on the fact that the functions \(S^{l,m}_n\) in App 5 are independent of any function parameters and injective, which I cannot find in any of the early texts (including Kleene [17]) even for \(m=0\).

  7. 7.

    Spector [48] eliminates dramatically the most obvious approach at limit ordinals: No increasing sequence \(\mathbf d _0<\mathbf d _1<\cdots \) of Turing degrees has a least upper bound. Of course, this was not known to Davis, Kleene and Mostowski when they wrote these early papers.

  8. 8.

    Kleene’s obtuse coding (the 3 and 5 in the definition) is motivated by the plans he and Church had to develop a general “constructive theory of ordinals ” beyond Cantor’s first and second number classes. They never got into this, but some (non-trivial and highly technical) results were proved by others, cf.  Kreider and Rogers [25], Putnam [44], Enderton-Putnam [7]. We will not cover this topic here.

  9. 9.

    O and \(\leqslant _O\) are defined by a (simultaneous) inductive definition as in App 10 which (in Kleene’s words) “is regarded from the finitary point of view as a correction, in that it eliminates the presupposition of the classical (non-constructive) second number class”. There are problems with this view, partly because many results about constructive ordinals cannot be proved (or even stated) without referring to ordinals. In any case, we will use \(\text {S}_1\) here.

  10. 10.

    A proof of this basic fact is included in §8 of Moschovakis [37].

  11. 11.

    For a discussion of the Spector Uniqueness Theorem and an outline of its proof for \(S_1\) see §9 of Moschovakis [37].

  12. 12.

    This strong uniqueness property cannot be extended to \(\omega ^2\), cf. Moschovakis [31], Nelson [41].

  13. 13.

    Choose \(\overline{k}\) such that \(\{\overline{k}\}(t,\varvec{x},\varvec{\alpha })=f(S^{1,m}_n(t,t),\varvec{x},\varvec{\alpha })\) and take \(e=S^{1,m}_n(\overline{k}, \overline{k})\).

  14. 14.

    In the terminology of Post [42], the proof shows that \(H_a\) and \(L_a\) are equivalent by bounded truth tables. Had Davis chosen to set \(L_1=\mathbb {N}\) at the basis, then these modified \(L_a\)s are recursively isomorphic with Kleene’s \(H_a\) sets, and by a simpler argument than the proof of this Lemma.

  15. 15.

    Cf. Moschovakis [36, 37] for a discussion (and many examples), and 7A.4 of Moschovakis [35] for a specific result which codifies many of the applications of effective grounded recursion in Descriptive Set Theory.

  16. 16.

    The interested reader may want to look at Moschovakis [36] where it was necessary to develop this generalized abstract nonsense in some detail.

  17. 17.

    For a classical example, consider the coding of recursive partial functions specified by the Normal Form Theorem in App 5. Its precise definition depends on the choice of computation model that we use, Turing machines, systems of recursive equations or whatever, but all these codings are equivalent and so uniform propositions about them are coding invariant. §4.3–§4.5 of Rogers [45] considers this situation in some detail and formulates stronger notions of equivalence than the one we use.

  18. 18.

    A pointclass in this paper is any collection \(\Gamma \) of relations \(P(\varvec{x},\varvec{\alpha })\) with arguments in \(\mathbb {N}\) and \(\mathscr {N}\). It is an awkward term but useful, and is has been well established since the 70 s for collections of relations in various spaces typically specified by the context.

  19. 19.

    They also suffice to prove that the notation system \(S_1\) is \(\varPi ^1_1\), cf.  Lemma 1 in the proof of Theorem 9.2 in Moschovakis [37].

  20. 20.

    What’s missing in their papers is the second part in the proof of the Boundedness Lemma 5.3.8 which looks tricky at first sight but is a standard, elementary tool in this area. It computes “witnesses to counterexamples” using diagonalization in very general circumstances, and we have already used it to establish the uniform properties of the jump in Footnote 6.

  21. 21.

    Cf. App 9 for the notation we use about wellorderings and ranks.

  22. 22.

    It is also his last paper on the subject.

  23. 23.

    We need to include all arithmetical sets in \(\varDelta (\mathscr {F})\), ow. \(\varDelta (\emptyset )=\emptyset \) and \(\varDelta \) would close at 0 and build up the empty set.

  24. 24.

    We assume some formal treatment of recursive substitutions into \(\mathsf A ^2\) formulas. In this case, the relevant recursive function is \((\alpha ,s)\mapsto (\alpha )_s\), and we use the equivalences

    $$ \varphi (s,(\alpha )_s,\gamma )\iff (\exists \delta )[\delta =(\alpha )_s{~ \& ~}\varphi (s,\delta ,\gamma )]\iff (\forall \delta ) [\delta =(\alpha )_s\rightarrow \varphi (s,\delta ,\gamma )]. $$

    These are satisfied by every model \(\mathscr {F}\) of \((\varDelta ^0_\infty \text {-Comp})\).

  25. 25.

    The converse fails, cf.  Steel [51].

  26. 26.

    This is not quite explicit in Spector [47], but Sacks [46] (8.5) credits it to Spector and I think this is right.

  27. 27.

    This is seriously implicit in §24 of Kleene [20], but the idea of the proof is there and Spector correctly credits Kleene for it.

  28. 28.

    The “1” is necessary here, in fact it is not the case that every \(\varPi ^1_1\) set is the least fixed point \(\overline{\varPhi }\) of an arithmetical monotone operator \(\varPhi \) on \(\mathbb {N}\), cf.  Feferman [8] and Moschovakis [34] (8.13, falsely claimed in the 1974 edition for all “countable acceptable structures”). Feferman’s result was the first applications of Cohen’s forcing to arithmetic.

  29. 29.

    Cf. Kechris and Moschovakis [12] for a relatively simple introduction to recursion in higher types and Sacks [46] for a full development.

  30. 30.

    Kleene [23] does not mention this and I recall him saying (much later) that he was not certain that the notion of a recursive partial function in higher type recursion was natural, but I cannot point to a reference for this.

References

  1. Addison, J. W. (1959). Separation principles in the hierarchies of classical and effective descriptive set theory. Fundamenta Methematicae, 46:123–135.

    Google Scholar 

  2. Church, A. (1935). An unsolvable problem in elementary number theory. Bulletin of the American Mathematical Society, 41:332–333. This is an abstract of [3].

    Google Scholar 

  3. Church, A. (1936). An unsolvable problem in elementary number theory. American Journal of Mathematics, pages 345–363. An abstract of this paper was published in [52].

    Google Scholar 

  4. Davis, M. (1950a). On the theory of recursive unsolvability. PhD thesis, Princeton University.

    Google Scholar 

  5. Davis, M. (1950b). Relatively recursive functions and the extended Kleene hierarchy. page 723. Proceedings of the International Congress of Mathematicians, Cambridge, Mass, 1950.

    Google Scholar 

  6. Davis, M. (1965). The undecidable. Raven Press.

    Google Scholar 

  7. Enderton, H. B., and Putnam, H. (1970). A note on the hyperarithmetical hierarchy. The Journal of Symbolic Logic, 35:429–430.

    Google Scholar 

  8. Feferman, S. (1965). Some applications of forcing and generic sets. Fundamenta Methematicae, 56:325–345.

    Google Scholar 

  9. Gandy, R., Kreisel, G., and Tait, W. (1960). Set existence. Bulletin of the Polish Academy of Sciences, Series in Science, Mathematics and Astronomy, 8:577–582.

    Google Scholar 

  10. Gandy, R. O. (1960). Proof of Mostowski’s conjecture. Bulletin de l’Academie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, 8:571–575.

    Google Scholar 

  11. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica and verwandter Systeme, I. Monatshefte für Mathematik und Physik, pages 173–198. English translations in [53] and [54].

    Google Scholar 

  12. Kechris, A. S. and Moschovakis, Y. N. (1977). Recursion in higher types. In Barwise, J., editor, Handbook of Mathematical Logic, Studies in Logic, No. 90, pages 681–737. North Holland, Amsterdam.

    Google Scholar 

  13. Kleene, S. C. (1936). General recursive functions of natural numbers. Mathematische Annalen, 112:727–742.

    Google Scholar 

  14. Kleene, S. C. (1938). On notation for ordinal numbers. Journal of Symbolic Logic, 3:150–155.

    Google Scholar 

  15. Kleene, S. C. (1943). Recursive predicates and quantifiers. Transactions of the American Mathematical Society, 53:41–73.

    Google Scholar 

  16. Kleene, S. C. (1950). A symmetric form of Gödel’s theorem. Konin. Neder. Akad. van Weten. te Amst. Proc. of the Section of Sciences, 53:800–802.

    Google Scholar 

  17. Kleene, S. C. (1952). Introduction to metamathematics. North Holland Co: D. Van Nostrand Co.

    Google Scholar 

  18. Kleene, S. C. (1953). Arithmetical predicates and function quantifiers. Journal of Symbolic Logic, 18:190. abstract of a talk presented at a meeting of the Association for Symbolic Logic on December 29, 1952.

    Google Scholar 

  19. Kleene, S. C. (1955a). åArithmetical predicates and function quantifiers. Transactions of the American Mathematical Society, 79:312–340.

    Google Scholar 

  20. Kleene, S. C. (1955b). On the form of predicates in the theory of constructive ordinals (second paper). American Journal of Mathematics, 77:405–428.

    Google Scholar 

  21. Kleene, S. C. (1955c). Hierarchies of number theoretic predicates. Bulletin of the American Mathematical Society, 61:193–213.

    Google Scholar 

  22. Kleene, S. C. (1959a). Quantification of number-theoretic functions. Compositio Mathematica, 14:23–40.

    Google Scholar 

  23. Kleene, S. C. (1959b). Recursive functionals and quantifiers of finite types I. Transactions of the American Mathematical Society, 91:1–52.

    Google Scholar 

  24. Kondo, M. (1938). Sur l’uniformization des complementaires analytiques et les ensembles projectifs de la second classe. Japanese Journal of Mathematics, 15:197–230.

    Google Scholar 

  25. Kreider, D. L., and Rogers, H, Jr. (1961). Constructive versions of ordinal number classes. Transactions of the American Mathematical Society, 100:325–369.

    Google Scholar 

  26. Kreisel, G. (1961). Set theoretic problems suggested by the notion of potential totality. Infinitistic methods (pp. 103–140). Pergamon, New York.

    Google Scholar 

  27. Kreisel, G. (1962). The axiom of choice and the class of hyperarithmetic functions. Indagationes Mathematicae, 24:307–319.

    Google Scholar 

  28. Lebesgue, H. (1905). Sur les fonctions represéntables analytiquement. Journal de Mathématiques 6 \(^{\rm {e}}\) serie, 1:139–216.

    Google Scholar 

  29. Markov, A. A. (1947). On the impossibility of certain algorithms in the theory of associative systems. Coptes rendus (Doklady) de l’Academie des Sciences de USSR, 55:583–586.

    Google Scholar 

  30. Markwald, W. (1954). Zur Theorie der konstruktiven Wohlordnungen. Mathematischen Annalen, 127:135–149.

    Google Scholar 

  31. Moschovakis, Y. N. (1966). Many-one degrees of the predicates \(H_a(x)\). Pacific Journal of Mathematics, 18:329–342.

    Google Scholar 

  32. Moschovakis, Y. N. (1968). Review of four papers on Church’s Thesis. The Journal of Symbolic Logic, 33:471–472.

    Google Scholar 

  33. Moschovakis, Y. N. (1969). Abstract first order computability II. Transactions of the American Mathematical Society, 138:465–504.

    Google Scholar 

  34. Moschovakis, Y. N. (1974). Elementary Induction on Abstract Structures. North Holland, Amsterdam. Studies in Logic, No. 77. Republished by Dover Publications, Mineola, NY, 2008, with a correction to 8.3.

    Google Scholar 

  35. Moschovakis, Y. N. (2009). Descriptive set theory, Second edition, volume 155 of Mathematical Surveys and Monographs. American Mathematical Society.

    Google Scholar 

  36. Moschovakis, Y. N. (2010a). Classical descriptive set theory as a refinement of effective descriptive set theory. Annals of Pure and Applied Logic, 162:243–255.

    Google Scholar 

  37. Moschovakis, Y. N. (2010b). Kleene’s amazing second recursion theorem. The Bulletin of Symbolic Logic, 16:189–239.

    Google Scholar 

  38. Mostowski, A. (1947). On definable sets of positive integers. Fundamenta Methematicae, 34:81–112.

    Google Scholar 

  39. Mostowski, A. (1951). A classification of logical systems. Studia Philosophica, 4:237–274.

    Google Scholar 

  40. Myhill, J. (1955). Creative sets. Zeitschrifft für Mathematische Logik und Grundlagen der Mathematik, 1:97–108.

    Google Scholar 

  41. Nelson, G. C. (1974). Many-one reducibility within the Turing degrees of the hyperarithmetic sets \(Ha(x)\). Transactions of the American Mathematical Society, 191:1–44.

    Google Scholar 

  42. Post, E. L. (1944). Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society, 50:284–316.

    Google Scholar 

  43. Post, E. L. (1947). Recursive unsolvability of a problem of Thue. The Journal of Symbolic Logic, 12:1–11.

    Google Scholar 

  44. Putnam, H. (1961). Uniqueness ordinals in higher constructive number classes. Essays on the foundations of mathematics, pages 190–206. Magnes Press, Hebrew University, Jerusalem.

    Google Scholar 

  45. Rogers, Jr., H. (1967). Theory of recursive functions and effective computability. McGraw-Hill.

    Google Scholar 

  46. Sacks, G. E. (1990). Higher Recursion Theory. Perspectives in Mathematical Logic: Springer.

    Google Scholar 

  47. Spector, C. (1955). Recursive wellorderings. Journal of Symbolic Logic, 20:151–163.

    Google Scholar 

  48. Spector, C. (1956). On degress of recursive unsolvability. Annals of Mathematics, 64:581–582.

    Google Scholar 

  49. Spector, C. (1960). Hyperarithmetical quantifiers. Fundamenta Methematicae, 48:313–320.

    Google Scholar 

  50. Spector, C. (1961). Inductively defined sets of natural numbers. Infinitistic methods, pages 97–102. Pergamon, New York.

    Google Scholar 

  51. Steel, J. R. (1978). Forcing with tagged trees. Annals of Mathematical Logic, 15:55–74.

    Google Scholar 

  52. Suslin, M. (1917). Sur une definition des ensembles measurables B sans nombres transfinis. Comptes Rendus Acad. Sci. Paris, 164:88–91.

    Google Scholar 

  53. Turing, A. M. (1936). On computable numbers with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42:230–265. A correction, ibid. volume 43 (1937), pp. 544–546.

    Google Scholar 

  54. Van Heijenoort, J., editor (1967). From Frege to Gödel, a source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge, Massachusetts, London, England.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, USA

    Yiannis N. Moschovakis

Authors
  1. Yiannis N. Moschovakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiannis N. Moschovakis .

Editor information

Editors and Affiliations

  1. University of Trieste, Trieste, Italy

    Eugenio G. Omodeo

  2. University of Udine, Udine, Italy

    Alberto Policriti

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Moschovakis, Y.N. (2016). Hyperarithmetical Sets. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation