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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

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

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