Finite-tower bounds for Skolem functions
Abstract
We bound the eventual order types of Skolem functions below finite exponential towers. Writing E0(u)=u, En+1(u)=2En(u), and ω0=1, ωk+1=ωωk, the argument gives |Sk<En(xm)|<ωrn, rn=2+(n(n+3))/2 (n≥1, m≥2 fixed). For triple towers we obtain the sharper bound |Sk<E3(xm)|<ω10 for every fixed m≥1. 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|=ε0.
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BibTeX
@misc{weiermann2026finitetower,
title = {{Finite-tower bounds for Skolem functions}},
author = {Andreas Weiermann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12740},
eprint = {2609.12740},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12740},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-tower bounds for Skolem functions
AU - Andreas Weiermann
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12740
UR - https://arxiv.org/abs/2609.12740
AB - 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, r_n=2+(n(n+3))/2 (n≥1, m≥2 fixed). For triple towers we obtain the sharper bound |Sk_<E_3(x^m)|<ω_10 for every fixed m≥1. 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|=ε_0.
ER -