Skolem函数的有限塔界

中文摘要 AI 辅助

我们对有限指数塔之下的Skolem函数的最终序型给出界。记$E_0(u)=u$，$E_{n+1}(u)=2^{E_n(u)}$，以及$\omega_0=1$，$\omega_{k+1}=\omega^{\omega_k}$，论证给出\\[ |\Sk_{<E_n(x^m)}|<\omega_{r_n},\qquad r_n=2+\frac{n(n+3)}2\quad(n\ge1,\\ m\ge2\text{ 固定}). \\] 对于三重塔，我们获得更精确的界$|\Sk_{<E_3(x^m)}|<\omega_{10}$，对每个固定的$m\ge1$成立。证明结合了渐近展开的比较与有限递归分解，以及对有序和与积的序型估计。解析部分在经典的对数-指数级数域中展开。我们在此设定下证明了所需的离散性、支撑和截断命题。外部输入是van den Dries--Macintyre--Marker的级数构造以及从Berarducci--Mamino回顾的序与序型估计。这些界蕴含$|\Sk|=\varepsilon_0$。

英文摘要

We bound the eventual order types of Skolem functions below finite exponential towers. Writing $E_0(u)=u$, $E_{n+1}(u)=2^{E_n(u)}$, and $ω_0=1$, $ω_{k+1}=ω^{ω_k}$, the argument gives \[ |\Sk_{<E_n(x^m)}|<ω_{r_n},\qquad r_n=2+\frac{n(n+3)}2\quad(n\ge1,\ m\ge2\text{ fixed}). \] For triple towers we obtain the sharper bound $|\Sk_{<E_3(x^m)}|<ω_{10}$ for every fixed $m\ge1$. The proof combines comparisons of asymptotic expansions with finite recursive decompositions and ordinal estimates for ordered sums and products. The analytic part is developed in the classical field of logarithmic-exponential series. We prove the required discreteness, support and truncation statements in this setting. The external inputs are the series construction of van den Dries--Macintyre--Marker and the order and ordinal estimates recalled from Berarducci--Mamino. These bounds imply $|\Sk|=\varepsilon_0$.

发表机构

• Ghent University（根特大学）

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