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 $\frac{x_1+\ldots+x_n+1}{n}$. 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 定义了 furable 数数的固定值 $\ mathb{R\\ t\ math} 美元。 它们显示, 通过反复应用函数生成的 $frac$, xx+y+1+2美元 。 Erickson、 nivasch 和 Xu 显示, $\ mathcal F$ 的订单类型是 $\ varepsil_ 0美元。 它们还调查了一个重现定义的函数 $M\ croomb{R\ t\ t\ mathb{R} 美元。 本文显示, 美元的不连续点数是 $massalfal $_ fal_ 美元 美元 。 它们还显示, 虽然 $malmoxal\ xxxxx 的总计函数是 $\ mal\ max mal\ max 普通函数 。