Home

# Recent Advances in Ordinal Analysis: Π12 — CA and Related Systems

Abstract

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of -analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to -formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated -comprehension, e.g., -comprehension. The details will be laid out in [28].

Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2 is still something to be sought. However, for reasons that cannot be explained here, -comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2 in the foreseeable future.

Roughly speaking, ordinally informative proof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results.

References
Hide All
[1] Barwise J., Admissible sets and structures, Springer-Verlag, Berlin, 1975.
[2] Buchholz W., Eine Erweiterung der Schnitteliminationsmethode, Habilitationsschrift, München, 1977.
[3] Buchholz W., A simplified version of local predicativity, Leeds proof theory 1991, Cambridge University Press, Cambridge, 1993, pp. 115147.
[4] Buchholz W., Feferman S., Pohlers W., and Sieg W., Iterated inductive definitions and subsystems of analysis, Springer-Verlag, Berlin, 1981.
[5] Buchholz W. and Schütte K., Proof theory of impredicative subsystems of analysis, Bibliopolis, Naples, 1988.
[6] Feferman S., Systems of predicative analysis, Journal of Symbolic Logic, vol. 29 (1964), pp. 130.
[7] Feferman S., A language and axioms for explicit mathematics, Lecture notes in mathematics 450, Springer-Verlag, Berlin, 1975, pp. 87139.
[8] Feferman S., Constructive theories of functions and classes, Logic colloquium '78, North-Holland, Amsterdam, 1979, pp. 159224.
[9] Feferman S., Proof theory: a personal report, Proof theory (Takeuti G., editor), North-Holland, Amsterdam, second ed., 1987, pp. 445485.
[10] Feferman S., Hilbert's program relativized: Proof-theoretical and foundational reductions, Journal of Symbolic Logic, vol. 53 (1988), pp. 364384.
[11] Feferman S., Remarks for “the trends in logic”, Logic colloquium '88, North-Holland, Amsterdam, 1989, pp. 361363.
[12] Friedman H., Robertson N., and Seymour P., The metamathematics of the graph minor theorem, Contemporary Mathematics, vol. 65 (1987), pp. 229261.
[13] Girard J.-Y., Introduction to –logic, Synthese, vol. 62 (1985), pp. 191216.
[14] Gödel K., The modern development of the foundations of mathematics in the light of philosophy, K. Gödel, Collected works, vol. III, Oxford University Press, Oxford, 1995, pp. 374387.
[15] Griffor E. and Rathjen M., The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347385.
[16] Hilbert D., Die Grundlegung der elementaren Zahlentheorie, Mathematische Annalen, vol. 104 (1931).
[17] Jäger G., Beweistheorie von KPN, Archiv für Mathematische Logik und Grundlagenforschung, vol. 20 (1980), pp. 5364.
[18] Jäger G., Zur Beweistheorie der Kripke-Platek Mengenlehre über den natürlichen Zahlen, Archiv für Mathematische Logik und Grundlagenforschung, vol. 22 (1982), pp. 121139.
[19] Jäger G., A well-ordering proof for Feferman s theory T0 , Archiv für Mathematische Logik und Grundlagenforschung, vol. 23 (1983), pp. 6577.
[20] Jäger G. and Pohlers W., Eine beweistheoretische Untersuchung von — CA + BI und verwandter Systeme, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch–Naturwissenschaftliche Klasse, 1982.
[21] Jensen R. B., The fine structure of the constructible hierarchy, Annals of Math. Logic, vol. 4 (1972), pp. 229308.
[22] Martin D. and Steel J., Projective determinacy, Proceedings of the National Academy of Sciences ofthe United States of America, vol. 85, 1988, pp. 65826586.
[23] Martin-Löf P., Intuitionistic type theory, Bibliopolis, Naples, 1984.
[24] Pohlers W., Proof-theoretical analysis of IDv by the method of localpredicativity, in [4], Springer, Berlin, 1981, pp. 261357.
[25] Pohlers W., Cut elimination for impredicative infinitary systems, part I: Ordinal analysis of ID1 , Archiv für Mathematische Logik und Grundlagenforschung, vol. 21 (1981), pp. 6987.
[26] Pohlers W., Cut elimination for impredicative infinitary systems, part II: Ordinal analysis for iterated inductive definitions, Archiv für Mathematische Logik und Grundlagenforschung, vol. 22 (1982), pp. 113129.
[27] Pohlers W., Proof theory and ordinal analysis, Archive for Mathematical Logic, vol. 30 (1991), pp. 311376.
[28] Rathjen M., An ordinal analysis of comprehension and related systems, in preparation.
[29] Rathjen M., Proof-theoretic analysis of KPM, Archive for Mathematical Logic, vol. 30 (1991), pp. 377403.
[30] Rathjen M., How to develop proof–theoretic ordinal f unctions on the basis of admissible sets, Mathematical Quarterly, vol. 39 (1993), pp. 4754.
[31] Rathjen M., Collapsing functions based on recursively large ordinals: A well–ordering proof for KPM, Archive for Mathematical Logic, vol. 33 (1994), pp. 3555.
[32] Rathjen M., Proof theory of reflection, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 181224.
[33] Richter W. and Aczel P., Inductive definitions and reflecting properties of admissible ordinals, Generalized recursion theory, North-Holland, Amsterdam, 1973, pp. 301381.
[34] Schlüter A., Provability in set theories with reflection, submitted.
[35] Schütte K., Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie, Mathematische Annalen, vol. 122 (1951), pp. 369389.
[36] Schütte K., Beweistheorie, Springer-Verlag, Berlin, 1960.
[37] Schütte K., Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik, Archiv für Mathematische Logik und Grundlagenforschung, vol. 67 (1964), pp. 4560.
[38] Schütte K., Predicative well-orderings, Formal systems and recursive functions, North-Holland, 1965, pp. 176184.
[39] Setzer T., Proof theoretical strength of Martin-löf type theory with w-type and one universe, Thesis, University of Munich, 1993.
[40] Simpson S., Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher Bäume, Archiv für Mathematische Logik und Grundlagenforschung, vol. 25 (1985), pp. 4565.
[41] Solovay R. M., Reinhardt W. N., and Kanamori A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.
[42] Takeuti G., Consistency proofs of subsystems of classical analysis, Annals of Mathematics, vol. 86 (1967), pp. 299348.
[43] Takeuti G., Proof theory and set theory, Synthese, vol. 62 (1985), pp. 255263.
[44] Takeuti G. and Yasugi M., The ordinals of the systems of second order arithmetic with the provably –comprehension and the –comprehension axiom respectively, Japanese Journal of Mathematics, vol. 41 (1973), pp. 167.
[45] Woodin H., Large cardinal axioms and independence: The continuum problem revisited, Mathematical Intelligencer, 1994, pp. 3135.
Recommend this journal

Bulletin of Symbolic Logic
• ISSN: 1079-8986
• EISSN: 1943-5894
• URL: /core/journals/bulletin-of-symbolic-logic
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 2 *