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Uniformization and skolem functions in the class of trees

Published online by Cambridge University Press:  12 March 2014

Shmuel Lifsches  and
Saharon Shelah
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
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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 definable Skolem functions (by a monadic formula with parameters)? This continues [6] where the question was asked only with respect to choice functions. A natural subclass is defined and proved to be the class of trees with definable Skolem functions. Along the way we investigate the spectrum of definable well orderings of well ordered chains.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 103 - 127
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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. 295–311.CrossRefGoogle Scholar
[2]Feferman, S. and Vaught, R. L., The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[3]Gurevich, Y., Monadic second-order theories, Model theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 479–506.Google Scholar
[4]Gurevich, Y. and Shelah, S., Rabin's uniformization problem, this Journal, vol. 48 (1983), pp. 1105–1119.Google Scholar
[5]Hausdorff, F., Grundzüge einer Theorie der geordnetn Mengen, Mathematische Annalen, vol. 65 (1908), pp. 435–505.CrossRefGoogle Scholar
[6]Lifsches, S. and Shelah, S., Uniformization, choice functions and well orders in the class of trees, this Journal, vol. 61 (1996), pp. 1206–1227.Google Scholar
[7]Rabin, M. O., Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 1–35.Google Scholar
[8]Shelah, S., The monadic theory of order, Annals of Mathematics. Second Series, vol. 102 (1975), pp. 379–419.CrossRefGoogle Scholar