Abstract
We survey a selection of results about majorization hierarchies. The main focus is on classical and recent results about the comparison between the slow and fast growing hierarchies.
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REFERENCES
Ackermann, W.: 1940, Zur Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen 117, 162–194.
Aczel, P.: 1980, Another Elementary Treatment of Girard's Result Connecting the Slow and Fast Growing Hierarchies of Number-Theoretic Functions, manuscript.
Arai, T.: 1993, ‘A Slow Growing Analogue to Buchholz’ Proof’, Annals of Pure and Applied Logic 54, 101–120.
Arai, T.: 1997, ‘Consistency Proof Via Pointwise Induction’, Archive for Mathematical Logic 37, 149–165.
Arai, T.: 1998, ‘Variations on a Theme by Weiermann’, The Journal of Symbolic Logic 63, 897–925.
Blankertz, B. and Weiermann, A.: 1996, ‘How to Characterize Provably Total Functions by the Buchholz Operator Method’, in Petr Hajek (ed.), Gödel '96, Logical Foundations of Mathematics, Computer Science and Physics — Kurt Gödel's Legacy, Proceedings of a conference, Brno, Czech Republic, August 1996, Berlin [Lecture Notes in Logic 6], pp. 205–213.
Buchholz, W.: 1980, Three Contributions to the Conference on Recent Advances in Proof Theory, Oxford, mimeographed.
Buchholz, W.: 1986, ‘A New System of Proof-Theoretic Ordinal Functions’, Annals of Pure and Applied Logic 32, 195–207.
Buchholz, W., E. A. Cichon and A. Weiermann: 1994, ‘A Uniform Approach to Fundamental Sequences and Hierarchies’, Mathematical Logic Quarterly 40, 273–286.
Cichon, E. A.: 1992, ‘Termination Proofs and Complexity Characterisations’, in Peter Aczel, Harold Simmons and Stanley Wainer (eds), Proof Theory, A Selection of Papers from the Leeds Proof Theory Programme, an International Summer School and Conference on Proof Theory, Leeds University, UK, 24 July–2 August 1990, Cambridge, pp. 173–193.
Cichon, E. A. and S. S. Wainer: 1983, ‘The Slow-Growing and the Grzegorczyk Hierarchies’, Journal of Symbolic Logic 48, 399–408.
Cichon, E. A. and A. Weiermann (1997), ‘Term Rewriting Theory for the Primitive Recursive Functions’, Annals of Pure and Applied Logic 83, 199–223.
Dershowitz, N. and M. Okada: 1988, ‘Proof-Theoretic Techniques for Term-Rewriting Theory’, in IEEE Computer Society (ed.), Proceedings of the Third Annual Symposium on Logic in Computer Science (LICS '88), Edinburgh, Scotland, UK, 5–8 July 1988, Edinburgh, pp. 104–111.
Feferman, S.: 1968, ‘Systems of Predicative Analysis, II: Representations of Ordinals’, Journal of Symbolic Logic 33, 193–220.
Friedman, H. and M. Sheard: 1995, ‘Elementary Descent Recursion and Proof Theory’, Annals of Pure and Applied Logic 71, 1–47.
Girard, J. Y.: 1981, ‘\(\Pi _2^1 \) Logic, Part 1', Annals of Mathematical Logic 21, 75–219.
Hardy, G.: 1904, ‘A Theorem Concerning the Infinite Cardinal Numbers’, Quarterly Journal of Mathematics 35, 87–94.
Grzegorczyk, A.: 1953,‘Some Classes of Recursive Functions’, Rozprawy Matematyczne 4.
Hofbauer, D.: 1992, ‘Termination Proofs by Multiset Path Orderings Imply Primitive Recursive Derivation Lengths’, Theoretical Computer Science 105, 129–140.
Howard, W. A.: 1970, ‘Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type’, in A. Kino, J. Myhill and R. Wesley (eds), Intuitionism and Proof Theory, Proceedings of the summer conference at Buffalo N.Y. 1968, Amsterdam [Studies in Logic and the Foundations of Mathematics 8], pp. 443–458.
Jervell, H. R.: 1979, Homogeneous Trees, Lecture Notes at the University of München.
Möllerfeld, M. and A. Weiermann: 1996, A Uniform Approach to ≺-Recursion, Münster, preprint.
Ritchie, D. M.: 1968, Program Structure and Computational Complexity, Doctoral dissertation. Harvard University.
Rose, H. E.: 1984, Subrecursion: Functions and Hierarchies, Clarendon Press, Oxford.
Schmerl, U. R.: 1981, Ñber die schwach und die stark wachsende Hierarchie zahlentheoretischer Funktionen, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse.
Schütte, K.: 1977, Proof Theory, Springer.
Schütte, K.: 1986/1987, ‘Majorisierungsrelationen und Fundamentalfolgen eines Ordinalzahlensystems von G. Jäger’, Archiv für Mathematische Logik 26, 29–55.
Schwichtenberg, H.: 1971, ‘Eine Klassifikation der ɛ0-rekursiven Funktionen’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 17, 61–74.
Schwichtenberg, H.: 1990, Homogeneous Trees and Subrecursive Hierarchies, Lecture at the University of München.
Vogel, H.: 1977, ‘Ausgezeichnete Folgen für prädikativ rekursive Ordinalzahlen und prädikativ rekursive Funktionen’, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 23, 435–438.
Wainer, S. S.: 1970, ‘A Subrecursive Hierarchy over the Predicative Ordinals’, in Wilfrid Hodges (ed.), Conference in Mathematical Logic, London, 1970, Berlin [Lecture Notes in Mathematics 255], pp. 350–351.
Wainer, S. S.: 1972, ‘Ordinal Recursion and a Refinement of the Ordinal Recursive Functions’, Journal of Symbolic Logic 37, 281–292.
Wainer, S. S.: 1989, ‘Slow Growing Versus Fast Growing’, Journal of Symbolic Logic 54 (1989), 608–614.
Weiermann, A.: 1995, ‘Termination Proofs by Lexicographic Path Orderings Imply Multiply Recursive Derivation Lengths’, Theoretical Computer Science 139, 355–362.
Weiermann, A.: 1996a, ‘Investigations on Slow Versus Fast Growing, Part I: How to Majorize Slow Growing Functions Nontrivially by Fast Growing Ones’, Archive for Mathematical Logic 34, 313–330.
Weiermann, A.: 1996b, ‘How to Characterize Provably Total Functions by Local Predicativity’, Journal of Symbolic Logic 61, 52–69.
Weiermann, A.: 1997, ‘Sometimes Slow Growing is Fast Growing’, Annals of Pure and Applied Logic 90, 91–99.
Weiermann, A.: 1998a, ‘How is it that Infinitary Methods can be Applied to Finitary Mathematics? Gödel's T: A Case Study’, Journal of Symbolic Logic 63, 1348–1370.
Weiermann, A.: 1998b, ‘Bounding Derivation Lengths with Functions from the Slow Growing Hierarchy’, Archive for Mathematical Logic 37, 427–441.
Weiermann, A.: 1999, ‘What Makes a (Pointwise) Hierarchy Slow Growing?’, in S. Barry Cooper and John K. Truss (eds), Sets and Proofs, Invited Papers from the Logic Colloquium 97, University of Leeds, England, 6–13 July 1997, Cambridge [London Mathematical Society Lecture Notes Series 258], pp. 403–423.
Weiermann, A.: 2001a, Γ0 May be Minimal Subrecursively Inaccessible, appeared in: Mathematical Logic Quarterly 47 (2001), 397–408.
Weiermann, A.: 2001b, ‘Some Interesting Connections between the Slow Growing Hierarchy and the Ackermann Function’, appeared in: Journal of Symbolic Logic 66 (2001), 609–628.
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Weiermann, A. Slow Versus Fast Growing. Synthese 133, 13–29 (2002). https://doi.org/10.1023/A:1020899506400
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DOI: https://doi.org/10.1023/A:1020899506400