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On the strength of Fraïssé’s conjecture

  • Richard A. Shore
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)

Abstract

We show that Fraïssé’s conjecture that the class of linear orderings is well-quasiordered under embedability is proof theoretically strong. Indeed, even special cases of its restriction to wellorderings implies ATR0: If the class of wellorderings has no infinite antichains or no infinite descending chains then ATR0. is provable in RCA0. These results answer questions of Clote, Friedman and Hirst.

Keywords

Initial Segment Rank Function Recursive Function Final Segment Recursive Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Aharoni, R., M. Magidor and R.A. Shore [1992], On the strength of König’s duality theorem for infinite bipartite graphs. Journal of Combinatorial Theory (B) 54, 257–290.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Chen, K.-H., Recursive well-founded orderings [1978], Annals of Mathematical Logic, 13, 117–147.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Clote, Peter [1990], The metamathematics of Fraïssé’s order type conjecture. In: Recursion Theory Week, K. Ambos-Spies, G.H. Müller, G.E. Sacks (eds.), LNMS 1432, 41–56. Springer-Verlag, Berlin.CrossRefGoogle Scholar
  4. van Englen, F., A.W. Miller and J. Steel [1987], Rigid Borel sets and better quasiorder theory. In Simpson [1987], 199–222.Google Scholar
  5. Fraïssé, R. [1948], Sur la comparison des types d’ordres. C.R. Acad. Sci. Paris, 226, 1330.MathSciNetzbMATHGoogle Scholar
  6. Friedman, H. [1975], Some systems of second order arithmetic and their use. In: Proceedings of the International Congress of Mathematicians (Vancouver 1974), 1, Canadian Mathematical Congress, 235–242.Google Scholar
  7. Friedman, H. and J. Hirst [1990], Weak comparability of well orderings and reverse mathematics. Annals of Pure and Applied Logic, 47, 11–29.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Friedman, H., N. Robertson and P.D. Seymour [1987], The metamathematics of the graph minor theorem. In Simpson [1987], 229–261.Google Scholar
  9. Harrington, L.A., M.D. Morley, A. Scedrov and S.G. Simpson (eds.) [1985], Harvey Friedman’s Research on the Foundations of Mathematics. North-Holland, Amsterdam.zbMATHGoogle Scholar
  10. Hirst, J. [1993], Reverse mathematics and ordinal exponentiation. To appear.Google Scholar
  11. Jockusch, C.G. Jr. and T.G. McLaughlin [1969], Countable retracing functions and П2 0 predicates. Pacific Journal of Mathematics, 30, 67–93.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Kruskal, J. [1960], Well-quasi-ordering, the tree theorem, and Vázsonyi’s conjecture. Transactions of the AMS, 95, 210–225.MathSciNetzbMATHGoogle Scholar
  13. Kunen, K. [1980], Set Theory, An Introduction to Independence Proofs. North-Holland, Amsterdam.zbMATHGoogle Scholar
  14. Laver, R. [1971], On Fraïssé’s order type conjecture. Annals of Mathematics, 93, 89–111.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Mansfield, R. and G. Weitkamp [1985], Recursive Aspects of Descriptive Set Theory. Oxford University Press, Oxford.zbMATHGoogle Scholar
  16. Metakides, G. and A. Nerode [1977], Recursively enumerable vector spaces. Annals of Mathematical Logic, 11, 147–171.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Metakides, G. and A. Nerode [1979], Effective content of field theory. Annals of Mathematical Logic, 17, 289–320.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Nash-Williams, C. St.J. A. [1968], On better-quasi-ordering transfinite sequences. Proc. of the Cambridge Philosophical Society, 64, 273–290.MathSciNetCrossRefGoogle Scholar
  19. Robertson, N. and P.D. Seymour [1990], Graph minors. IV. Tree-width and well-quasi-ordering. Journal of Combinatorial Theory, Series (B), 48, 227–254.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Robertson, N. and P.D. Seymour [1990], Graph minors, VIII. A Kuratowski theorem for general surfaces. Journal of Combinatorial Theory, Series (B), 48, 255–288.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Rogers, H. Jr. [1967], Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York.zbMATHGoogle Scholar
  22. Simpson, S.G. [1982], Σ1 1 and П1 1 transfinite induction. In: Logic Colloquium’ 80, D. van Dalen, D. Lascar and J. Smiley, eds., North-Holland, Amsterdam, 239–253.Google Scholar
  23. Simpson, S.G. [1985], BQO theory and Fraïssé’s conjecture. Chapter 9 of Mansfield and Weitkamp [1985].Google Scholar
  24. Simpson, S.G. [1985a], Friedman’s research on subsystems of analysis. In Harrington et al. [1985], 137–160.Google Scholar
  25. Simpson, S.G. (ed.) [1987], Logic and Combinatorics. Contemporary Mathematics, 65, American Mathematical Society, Providence.zbMATHGoogle Scholar
  26. Simpson, S.G. [1993], Subsystems of Second Order Arithmetic. Springer-Verlag, Berlin.Google Scholar
  27. Spector, C. [1958], Strongly invariant hierarchies. Abstract, Notices of the AMS, 5, 851.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Richard A. Shore
    • 1
  1. 1.Cornell UniversityUSA

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