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 x+y+12. Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε0. 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 ε0. 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+12 to an n-ary function is the function x1+⋯+xn+1n. We show that this function generates a set Fn whose order type is just φn−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 φn−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.