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 $\varphi_{n-1}(0)$. 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 $\varphi_{n-1}(0)$. 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.

翻译：Erickson将可熔数定义为由函数 \(\frac{x+y+1}{2}\) 重复应用生成的实数集合 \(\mathcal F\)。Erickson、Nivasch 和 Xu 证明了 \(\mathcal F\) 是良序集，其序型为 \(\varepsilon_0\)。他们还研究了一个递归定义的函数 \(M\colon \mathbb{R}\to\mathbb{R}\)，证明 \(M\) 的不连续点集是 \(\mathcal F\) 的序型为 \(\varepsilon_0\) 的子集，并且尽管 \(M\) 是 \(\mathbb{R}\) 上的全函数，但 \(M\) 在 \(\mathbb{Q}\) 上的限制为全函数这一事实却无法在一阶皮亚诺算术 \(\mathsf{PA}\) 中证明。本文探讨了 Friedman 提出的问题：类似方法能否生成序型更大的良序集 \(\mathcal F\)？正如 Friedman 所指出的，Kruskal 树定理给出了由单调函数 \(g:\mathbb R^n\to\mathbb R\) 重复应用生成的任意集合的序型的上界为小 Veblen 序数。将 \(\frac{x+y+1}{2}\) 推广到 \(n\) 元函数的最直接方式是函数 \(\frac{x_1+\cdots+x_n+1}{n}\)。我们证明该函数生成的集合 \(\mathcal F_n\) 的序型恰为 \(\varphi_{n-1}(0)\)。为此，我们发展了递归定义的函数 \(M_n\colon \mathbb{R}\to\mathbb{R}\)，自然推广了函数 \(M\)。此外，我们证明对于任意线性函数 \(g:\mathbb R^n\to\mathbb R\)，生成的 \(\mathcal F\) 的序型至多为 \(\varphi_{n-1}(0)\)。最后，我们表明确实存在连续函数 \(g:\mathbb R^n\to\mathbb R\)，使得生成集合 \(\mathcal F\) 的序型趋近于小 Veblen 序数。