Journal of Symbolic Logic 48 (4):1105-1119 (1983)

Abstract
The set of all words in the alphabet {l, r} forms the full binary tree T. If x ∈ T then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of T and over arbitrary subsets of T. We prove that there is no monadic second-order formula φ * (X, y) such that for every nonempty subset X of T there is a unique y ∈ X that satisfies φ * (X, y) in T
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DOI 10.2307/2273673
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