Year: 2026
Type: preprint
Abstract: We investigate the first-order theories of full \(n\)-branching ordinal trees \(\mathfrak{T}_α^n\). We obtain a complete classification of the standard trees up to elementary equivalence: every \(\mathfrak{T}_α^n\) is elementarily equivalent to one of four canonical types determined by the ordinal \(α\). For each canonical type we establish effective quantifier elimination and prove decidability of the theory. Along the way we develop the realizability theory of colored ordinal characters and obtain a sharp bound on the length of minimal witnesses. We also clarify the relationship between these trees and monadic second-order logic over ordinals, and show that the equal-height relation is not first-order definable in any standard tree.
Source: arXiv (Cornell University)
Authors: Chen, Junhong, Zhang, Yi
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202212913