论文标题
通用的熔断数及其顺序
Generalized fusible numbers and their ordinals
论文作者
论文摘要
埃里克森（Erickson）通过重复应用$ \ frac {x+y+1} {2} $重复应用程序生成的真实数字定义了fusible数字。 Erickson，Nivasch和Xu表明$ \ Mathcal F $的订购良好，订单类型$ \ VAREPSILON_0 $。他们还研究了一个递归定义的函数$ m \ colon \ mathbb {r} \ to \ mathbb {r} $。他们表明，$ m $的不连续点的点集是订单类型$ \ varepsilon_0 $的$ \ mathcal f $的子集。他们还表明，尽管$ m $是$ \ mathbb r $的总功能，但$ m $ to $ \ mathbb {q} $的限制在一阶peano arithmetic $ \ mathsf {pa} $中都无法证明。 在本文中，我们探讨了类似方法是否可以产生较大订单类型的$ \数学f $的问题（由弗里德曼提出）。正如弗里德曼（Friedman）指出的那样，克鲁斯卡尔（Kruskal）的树定理通过重复应用单调函数$ g：\ mathbb r^n \ to \ mathbb r $重复应用单调函数$ g：\ mathbb r^n \。 $ \ frac {x+y+1} {2} $对$ n $ -ary函数的最直接概括是函数$ \ frac {x_1+\ cdots+x_n+1} {n} $。我们表明，此功能生成了一个$ \ MATHCAL F_N $，其订单类型仅为$φ_{N-1}（0）$。为此，我们开发了递归定义的函数$ M_N \ COLON \ MATHBB {r} \ to \ MATHBB {R} $自然地概括了功能$ M $。 此外，我们证明，对于任何线性函数$ g：\ mathbb r^n \ to \ mathbb r $，生成的$ \ mathcal f $的订单类型最多是$φ_{n-1}（0）$。 最后，我们证明确实存在连续函数$ g：\ mathbb r^n \ to \ mathbb r $，为此，所得的订单类型$ \ mathcal f $接近小veblen ordinal。
Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$. They also investigated a recursively defined function $M\colon \mathbb{R}\to\mathbb{R}$. They showed that the set of points of discontinuity of $M$ is a subset of $\mathcal F$ of order type $\varepsilon_0$. They also showed that, although $M$ is a total function on $\mathbb R$, the fact that the restriction of $M$ to $\mathbb{Q}$ is total is not provable in first-order Peano arithmetic $\mathsf{PA}$. In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $\mathcal F$ of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function $g:\mathbb R^n\to\mathbb R$. The most straightforward generalization of $\frac{x+y+1}{2}$ to an $n$-ary function is the function $\frac{x_1+\cdots+x_n+1}{n}$. We show that this function generates a set $\mathcal F_n$ whose order type is just $φ_{n-1}(0)$. For this, we develop recursively defined functions $M_n\colon \mathbb{R}\to\mathbb{R}$ naturally generalizing the function $M$. Furthermore, we prove that for any linear function $g:\mathbb R^n\to\mathbb R$, the order type of the resulting $\mathcal F$ is at most $φ_{n-1}(0)$. Finally, we show that there do exist continuous functions $g:\mathbb R^n\to\mathbb R$ for which the order types of the resulting sets $\mathcal F$ approach the small Veblen ordinal.
