带后继函数的完全n分支序数树的初等理论
The elementary theory of full $n$-branching ordinal trees with successor functions
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中文总结 AI 辅助
该论文研究带后继函数的完全n分支序数树的初等理论，对标准树按初等等价分类，建立量词消去与可判定性结果，还发展可实现性理论并澄清树与一元二阶逻辑的关系。
中文摘要 AI 辅助
我们研究完全n分支序数树𝔗^n_α的一阶理论，对标准树按初等等价进行完整分类：每个𝔗^n_α初等等价于由序数α确定的四种典型类型之一。针对每种典型类型，我们建立了有效的量词消去法并证明该理论的可判定性。过程中，我们发展了有色序数特征的可实现性理论，得到了极小见证长度的精确界，还阐明了这些树与序数上的一元二阶逻辑的关系，证明等高层关系在任意标准树中都不是一阶可定义的。
英文摘要
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.
