First-order aspects of tree paths

Abstract

Tree paths are investigated using first-order logic. The following results are obtained: (i) every definable path can be defined by a firstorder formula using at most one parameter chosen from the path itself; (ii) a canonical representation of the formulas that define definable paths is obtained; and (iii) every tree that has only finitely many paths that are not definable is n-equivalent to a tree of which all paths are definable. Moreover, a certain property that might be expected to hold, involving the transfer of n-equivalence between trees, is shown not to be true.

First-order theories of bounded trees

“…This observation can be combined with an additive version of Ramsey's Theorem (see e.g. [She75]) to give Fact 28 below (refer also to the motivation of a certain partition property that is discussed in [Kel15], for a more detailed justification of Fact 28).…”