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Theories and Ordinals: Ordinal Analysis

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Book cover Computation and Logic in the Real World (CiE 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4497))

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Abstract

How do ordinals gauge the strength and computational power of theories and what kind of information can be extracted from this correlation? This will be the guiding question of this talk. The connection between ordinal representation systems and theories is established in ordinal analysis, a central area of proof theory. The origins of proof theory can be traced back to the second problem on Hilbert’s famous list of problems, which called for a proof of consistency of the arithmetical axioms of the reals.

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References

  1. Ackermann, W.: Begründung des, tertium non datur mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Dissertation, Göttingen (1924)

    Google Scholar 

  2. Arai, T.: Wellfoundedness proofs by means of non-monotonic inductive definitions. I. \(\Pi^0_2\)-operators. Journal of Symbolic Logic 69, 830–850 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arai, T.: Proof theory for theories of ordinals. II. Π 3-reflection. Annals of Pure and Applied Logic 129, 39–92 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bachmann, H.: Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordinalzahlen. Vierteljahresschrift Naturforsch. Ges. Zürich 95, 115–147 (1950)

    MATH  Google Scholar 

  5. Carlson, T.: Elementary patterns of resemblance. Annals of Pure and Applied Logic 108, 19–77 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Feferman, S.: Systems of predicative analysis II. Representations of ordinals. Journal of Symbolic Logic 33, 193–220 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feferman, S.: Proof theory: a personal report. In: Takeuti, G. (ed.) Proof Theory, 2nd edn, pp. 445–485. North-Holland, Amsterdam (1987)

    Google Scholar 

  8. Feferman, S.: Three conceptual problems that bug me. Unpublished lecture text for 7th Scandinavian Logic Symposium, 1996 (Uppsala)

    Google Scholar 

  9. Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen 112, 493–565 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  10. Girard, J.-Y.: A survey of \(\Pi^1_2\)-logic. Part I: Dilators. Annals of Mathematical Logic 21, 75–219 (1981)

    Article  MathSciNet  Google Scholar 

  11. Girard, J.-Y., Vauzeilles, J.: Functors and Ordinal Notations. I: A Functorial Construction of the Veblen Hierarchy. Journal of Symbolic Logic 49, 713–729 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Girard, J.-Y., Vauzeilles, J.: Functors and Ordinal Notations. II: A Functorial Construction of the Bachmann Hierarchy. Journal of Symbolic Logic 49, 1079–1114 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kanamori, A.: The higher infinite. Springer, Berlin (1995)

    Google Scholar 

  14. Kreisel, G.: A survey of proof theory. Journal of Symbolic Logic 33, 321–388 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  15. von Neumann, J.: Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift pp. 1–46 (1926)

    Google Scholar 

  16. Pohlers, W.: Proof theory and ordinal analysis. Archive for Mathematical Logic 30, 311–376 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pohlers, W.: Subsystems of set theory and second order number theory. In: Buss, S. (ed.) Handbook of proof theory, pp. 209–335. North-Holland, Amsterdam (1998)

    Chapter  Google Scholar 

  18. Rathjen, M.: Ordinal notations based on a weakly Mahlo cardinal. Archive for Mathematical Logic 29, 249–263 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rathjen, M.: Proof-Theoretic Analysis of KPM. Archive for Mathematical Logic 30, 377–403 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rathjen, M.: How to develop proof–theoretic ordinal functions on the basis of admissible sets. Mathematical Quarterly 39, 47–54 (1993)

    MATH  Google Scholar 

  21. Rathjen, M.: Collapsing functions based on recursively large ordinals: A well–ordering proof for KPM. Archive for Mathematical Logic 33, 35–55 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rathjen, M.: Proof theory of reflection. Annals of Pure and Applied Logic 68, 181–224 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rathjen, M.: Recent advances in ordinal analysis: \(\Pi^1_2\)-CA and related systems. Bulletin of Symbolic Logic 1, 468–485 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rathjen, M.: The realm of ordinal analysis. In: Cooper, S.B., Truss, J.K. (eds.) Sets and Proofs, pp. 219–279. Cambridge University Press, Cambridge (1999)

    Chapter  Google Scholar 

  25. Rathjen, M.: An ordinal analysis of stability. Archive for Mathematical Logic 44, 1–62 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rathjen, M.: An ordinal analysis of parameter-free \(\Pi^1_2\) comprehension. Archive for Mathematical Logic 44, 263–362 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Weiermann, A.: A Functorial Property of the Aczel-Buchholz-Feferman Function. Journal of Symbolic Logic 59, 945–955 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Woodin, W.H.: Large cardinal axioms and independence: The continuum problem revisited. The Mathematical Intelligencer 16(3), 31–35 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK

    Michael Rathjen

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  1. Michael Rathjen
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Editor information

Editors and Affiliations

  1. Dept. of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    S. Barry Cooper

  2. Institue for Logic, Language and Computation (ILLC), Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, TV Amsterdam, The Netherlands

    Benedikt Löwe

  3. Dipartimento di Scienze Matematiche ed Informatiche “R. Magari”, University of Siena, 53100, Siena, Italy

    Andrea Sorbi

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Rathjen, M. (2007). Theories and Ordinals: Ordinal Analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_65

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  • DOI: https://doi.org/10.1007/978-3-540-73001-9_65

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73000-2

  • Online ISBN: 978-3-540-73001-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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