Generalized fusible numbers and their ordinals

Erickson defined the fusible numbers as a set F of reals generated by repeated application of the function 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-+ 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 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 : Rn-+ R. The most straightforward generalization of x+y+1 2 to an n-ary function is the function x1+center dot center dot center dot+xn+1 . We show that this function generates a set Fn whose order n type is just phi n-1(0). For this, we develop recursively defined functions Mn : R-+R naturally generalizing the function M. Furthermore, we prove that for any linear function g : Rn-+ R, the order type of the resulting F is at most phi n-1(0). Finally, we show that there do exist continuous functions g : Rn-+ R for which the order types of the resulting sets F approach the small Veblen ordinal.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper investigates the properties of fusible numbers and recursively defined functions, and explores the limitations of the order types of well-ordered sets generated by repeated application of functions.