Generalized fusible numbers and their ordinals

TL;DR

This paper generalizes fusible numbers to higher dimensions, establishing their order types as functions of the Veblen hierarchy, and explores the impact of different functions on the complexity of generated well-ordered sets.

Contribution

It introduces a family of generalized fusible sets with explicitly determined order types and analyzes how various functions influence their ordinal complexity.

Findings

Generalized sets have order type 4a0; 4a0; 4a0;

Linear functions produce sets with order type at most 4a0;

Certain continuous functions can approach the small Veblen ordinal.

Abstract

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.