Generalized fusible numbers and their ordinals

Abstract

Erickson defined the fusible numbers as a set $F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $F$ is well ordered, with order type ${\epsilon }_0$. They also investigated a recursively defined function $M:R\to R$. They showed that the set of points of discontinuity of M is a subset of $F$ of order type ${\epsilon }_0$. They also showed that, although M is a total function on $R$, the fact that the restriction of M to $Q$ is total is not provable in first-order Peano arithmetic $\mathrm{PA}$.

In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $F$ 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 $g:R^n\to R$.

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 $F_n$ whose order type is just ${\phi }_{n-1}\left(0\right)$. For this, we develop recursively defined functions $M_n:R\to R$ naturally generalizing the function M.

Furthermore, we prove that for any linear function $g:R^n\to R$, the order type of the resulting $F$ is at most ${\phi }_{n-1}\left(0\right)$.

Finally, we show that there do exist continuous functions $g:R^n\to R$ for which the order types of the resulting sets $F$ approach the small Veblen ordinal.