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Uniformization, choice functions and well orders in the class of trees

Published online by Cambridge University Press:  12 March 2014

Shmuel Lifsches
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: sam@math.huji.ac.il
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il
Corresponding

Abstract

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the first class don't have a definable choice function and the trees in the second class have even a definable well ordering of their elements. This has a close connection to the uniformization problem.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1996

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References

[1]Büchi, J. R. and Landweber, L. H., Solving sequential conditions by finite-state strategies, Transactions of the American Mathematical Society, vol. 138 (1969), pp. 295311.CrossRefGoogle Scholar
[2]Gurevich, Y., Monadic second-order theories, Model theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 479506.Google Scholar
[3]Gurevich, Y. and Shelah, S., Rabin's uniformization problem, this Journal, vol. 48 (1983), pp. 11051119.Google Scholar
[4]Hausdorff, F., Grundzüge einer Theorie der geordnetn Mengen, Mathematische Annalen, vol. 65 (1908), pp. 435505.CrossRefGoogle Scholar
[5]Lifsches, S. and Shelah, S., Uniformization and Skolem functions in the class of trees, this Journal, accepted.Google Scholar
[6]Rabin, M. O., Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 135.Google Scholar
[7]Shelah, S., The monadic theory of order, Annals of Mathematics, vol. 102 (1975), pp. 379419.CrossRefGoogle Scholar

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