Abstract
Szpilrajn’s Theorem states that any partial orderP=〈S,<p〉 has a linear extensionP=〈S,<L〉. This is a central result in the theory of partial orderings, allowing one to define, for instance, the dimension of a partial ordering. It is now natural to ask questions like “Does a well-partial ordering always have a well-ordered linear extension?” Variations of Szpilrajn’s Theorem state, for various (but not for all) linear order typesτ, that ifP does not contain a subchain of order typeτ, then we can chooseL so thatL also does not contain a subchain of order typeτ. In particular, a well-partial ordering always has a well-ordered extension.
We show that several effective versions of variations of Szpilrajn’s Theorem fail, and use this to narrow down their proof-theoretic strength in the spirit of reverse mathematics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Bonnet,Stratifications et extension des genres de chaînes dénombrables, Comptes Rendus de l’Académie des Sciences, Paris, Série A-B269 (1969), A880-A882.
R. Bonnet and M. Pouzet,Extension et stratification d’ensembles dispersés, Comptes Rendus de l’Académie des Sciences, Paris, Série A-B268 (1969), A1512-A1515.
R. Bonnet and M. Pouzet,Linear extensions of ordered sets, inOrdered Sets (I. Rival, ed.), Proceedings of a NATO Advanced Study Institute held in Banff, Alta., August 28–September 12, 1981, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1982, pp. 125–170.
M. Dehn,Über unendliche diskontinuierliche Gruppen, Mathematische Annalen71 (1912), 116–144.
R. G. Downey,Computability theory and linear orderings, inHandbook of Recursive Mathematics, Vol. 2, North-Holland, Amsterdam, 1998, pp. 823–976.
Yu. L. Ershov, S. S. Goncharov, A. Nerode and J. Remmel (eds.),Handbook of Recursive Mathematics, Vols. 1 and 2, North-Holland, Amsterdam, 1998.
H. Friedman,Some systems of second order arithmetic and their uses, inProceedings of the International Congress of Mathematicians, Vancouver, 1974, Canadian Mathematical Congress, Montreal, Que., 1975, pp. 235–242.
H. Friedman, N. Robertson and P. Seymour,The metamathematics of the graph minor theorem, inLogic and Combinatorics (S. G. Simpson, ed.), American Mathematical Society, Providence, R.I., 1987, pp. 229–261.
H. Friedman, S. G. Simpson and R. L. Smith,Countable algebra and set existence axioms, Annals of Pure and Applied Logic25 (1983), 141–181.
A. Fröhlich and J. C. Shepherdson,Effective procedures in field theory, Transactions of the Royal Society of London, Series A248 (1956), 407–432.
C. G. Jockusch, Jr.,Ramsey’s theorem and recursion theory, Journal of Symbolic Logic37 (1972), 268–280.
C. G. Jockusch, Jr. and R. I. Soare, Π 01 classes and degrees of theories, Transactions of the American Mathematical Society173 (1972), 33–56.
P. Jullien,Contribution à l’étude des types d’ordre dispersés, Thèse, Marseille, 1969.
G. Metakides and A. Nerode,Effective content of field theory, Annals of Mathematical Logic17 (1979), 289–320.
J. Paris and L. Harrington,A mathematical incompleteness in Peano arithmetic, inHandbook of Mathematical Logic (J. Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 1133–1142.
M. O. Rabin,Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society95 (1960), 341–360.
H. G. Rogers, Jr.,Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.
J. G. Rosenstein,Linear Orderings, Academic Press, New York-London, 1982.
J. G. Rosenstein,Recursive linear orderings, inOrders: Description and Roles (Proceedings of the Fourth Conference on Ordered Sets and Their Applications held in L’Arbresle, July 5–11, 1982) (M. Pouzet and D. Richard, eds.), North-Holland, Amsterdam-New York, 1984, pp. 465–475.
S. G. Simpson,Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, 1999.
T. A. Slaman and W. H. Woodin,Extending partial orders to dense linear orders, Annals of Pure and Applied Logic94 (1998), 253–261.
R. I. Soare,Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin-New York, 1987.
R. Solomon,Ordered groups: a case study in reverse mathematics, Bulletin of Symbolic Logic5 (1999), 45–58.
E. Szpilrajn,Sur l’extension de l’ordre partiel, Fundamenta Mathematicae16 (1930), 386–389.
Author information
Authors and Affiliations
Additional information
This research was carried out while the first and second authors were visiting the third and fourth authors at the University of Wisconsin. The first and second authors’ research was partially supported by the Marsden Fund of New Zealand. The third author’s research is partially supported by NSF grant DMS-9732526. The fourth author is supported by an NSF postdoctoral fellowship.
Rights and permissions
About this article
Cite this article
Downey, R.G., Hirschfeldt, D.R., Lempp, S. et al. Computability-theoretic and proof-theoretic aspects of partial and linear orderings. Isr. J. Math. 138, 271–289 (2003). https://doi.org/10.1007/BF02783429
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02783429