自然井序的一元第二order limit定律
Monadic second order limit laws for natural well orderings
摘要：
通过结合Büchi的经典结果、一些基本的Tauberian定理以及逻辑学和组合学的一些基本工具，我们证明了每个带有$\varepsilon_0\geq α\geq ω^ω$的序数$α$满足自然一元二阶极限定律，并且每个带有$ω^ω>α\geq ω$的序数$α$满足自然一元二阶Cesaro极限定律。在这两种情况下，我们通常将$α$与子结构类$\{β:β<α\}$等同起来。我们工作在一个加法设置的范数函数$N$分配给每个序数$α$的符号$ω$在其康托规范形式的出现次数。这个数与树中与$α$规范相关的边数相同。对于一个给定的$α$与$ω\leq α\leq \varepsilon_0$的一元二阶公式$φ$的渐近概率是$\lim_{n\to\infty} \frac{\#\{β<α: Nβ=n\wedge β\models Φ\}}{\#\{β<α: Nβ=n\}}$，如果这个限制存在。如果这个极限只存在于Cesaro意义上，我们就称之为$φ$的Cesaro渐近概率。此外，我们证明了一元二阶极限定律的序数段以下$Γ_0$（其中的规范功能适当扩展），我们表明本文的结果可以扩展到更大的序数段，甚至某些impredicative序数符号系统具有符号的不可数序数。我们还简要地说明如何证明相应的乘法结果的设置定义相对于Matula编码。本文关于序数不超过$\varepsilon_0$的结果是与Alan R.伍兹.
英文摘要：
By combining classical results of Büchi, some elementary Tauberian theorems and some basic tools from logic and combinatorics we show that every ordinal $α$ with $\varepsilon_0\geq α\geq ω^ω$ satisfies a natural monadic second order limit law and that every ordinal $α$ with $ω^ω>α\geq ω$ satisfies a natural monadic second order Cesaro limit law. In both cases we identify as usual $α$ with the class of substructures $\{β:β<α\}$. We work in an additive setting where the norm function $N$ assigns to every ordinal $α$ the number of occurrrences of the symbol $ω$ in its Cantor normal form. This number is the same as the number of edges in the tree which is canonically associated with $α$. For a given $α$ with $ω\leq α\leq \varepsilon_0$ the asymptotic probability of a monadic second order formula $φ$ from the language of linear orders is $\lim_{n\to\infty} \frac{\#\{β<α: Nβ=n\wedge β\models Φ\}}{\#\{β<α: Nβ=n\}}$ if this limit exists. If this limit exists only in the Cesaro sense we speak of the Cesaro asympotic probability of $φ$. Moreover we prove monadic second order limit laws for the ordinal segments below below $Γ_0$ (where the norm function is extended appropriately) and we indicate how this paper's results can be extended to larger ordinal segments and even to certain impredicative ordinal notation systems having notations for uncountable ordinals. We also briefly indicate how to prove the corresponding multiplicative results for which the setting is defined relative to the Matula coding. The results of this paper concerning ordinals not exceeding $\varepsilon_0$ have been obtained partly in joint work with Alan R. Woods.
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