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Well-partial-orderings and the big Veblen number

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In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with the classical tree-embeddability, but in this article we are interested in more general situations (transfinite addresses and well-partially ordered addresses). We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal \({\vartheta \Omega^{\Omega}}\). Somewhat surprisingly, changing a little bit the well-partially ordered addresses (going from multisets to finite sequences), the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely \({\vartheta \Omega^{\Omega^\Omega}}\). Our results contribute to the research program (originally initiated by Diana Schmidt) on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings.

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References

  1. Kruskal, J.B.: The theory of well-quasi-ordering: a frequently discovered concept. J. Combin. Theory Ser. A 13, 297–305 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Schmidt, D.: Well-Partial Orderings and Their Maximal Order Types. Habilitationsschrift, Heidelberg (1979)

    Google Scholar 

  3. Simpson, S.G.: Nonprovability of certain combinatorial properties of finite trees. In: Harvey Friedman’s research on the foundations of mathematics, Stud. Logic Found. Math., vol. 117, pp. 87–117. North-Holland, Amsterdam (1985)

  4. Rathjen, M., Weiermann, A.: Proof-theoretic investigations on Kruskal’s theorem. Ann. Pure Appl. Logic 60(1), 49–88 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Schmidt, D.: Bounds for the closure ordinals of replete monotonic increasing functions. J. Symb. Logic 40(3), 305–316 (1975)

    Article  MATH  Google Scholar 

  6. Weiermann, A.: An order-theoretic characterization of the Schütte–Veblen-hierarchy. Math. Logic Q. 39(3), 367–383 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Schütte, K.: Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen. Math. Ann. 127, 15–32 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  8. Veblen, O.: Continuous increasing functions of finite and transfinite ordinals. Trans. Am. Math. Soc. 9(3), 280–292 (1908)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kříž, I.: Well-quasiordering finite trees with gap-condition. Proof of Harvey Friedman’s conjecture. Ann. Math. (2) 130(1), 215–226 (1989)

    Article  MATH  Google Scholar 

  10. Gordeev, L.: Generalizations of the Kruskal–Friedman theorems. J. Symb. Logic 55(1), 157–181 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. de Jongh, D.H.J., Parikh, R.: Well-partial orderings and hierarchies. Nederl. Akad. Wetensch. Proc. Ser. A 80 = Indag. Math. 39(3), 195–207 (1977)

    Article  MathSciNet  Google Scholar 

  12. Aschenbrenner, M., Pong, W.Y.: Orderings of monomial ideals. Fund. Math. 181(1), 27–74 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Weiermann, A.: Proving termination for term rewriting systems. In: Computer Science Logic (Berne, 1991), Lecture Notes in Computer Science, vol. 626, pp. 419–428. Springer, Berlin (1992)

  14. Weiermann, A.: A computation of the maximal order type of the term ordering on finite multisets. In: Mathematical Theory and Computational Practice, Lecture Notes in Computer Science, vol. 5635, pp. 488–498. Springer, Berlin (2009)

  15. Buchholz, W., Schütte, K.: Proof Theory of Impredicative Subsystems of Analysis, Studies in Proof Theory. Monographs, vol. 2. Bibliopolis, Naples (1988)

  16. Weiermann, A.: Complexity bounds for some finite forms of Kruskal’s theorem. J. Symb. Comput. 18(5), 463–488 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Jeroen Van der Meeren.

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Van der Meeren, J., Rathjen, M. & Weiermann, A. Well-partial-orderings and the big Veblen number. Arch. Math. Logic 54, 193–230 (2015). https://doi.org/10.1007/s00153-014-0408-5

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