Generalized fusible numbers and their ordinals
share
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 ε_0. They also investigated a recursively defined function Mℝ→ℝ. They showed that the set of points of discontinuity of M is a subset of ℱ of order type ε_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:ℝ^n→ℝ. The most straightforward generalization of x+y+1/2 to an n-ary function is the function x_1+…+x_n+1/n. We show that this function generates a set ℱ_n whose order type is just φ_n-1(0). For this, we develop recursively defined functions M_nℝ→ℝ naturally generalizing the function M. Furthermore, we prove that for any linear function g:ℝ^n→ℝ, the order type of the resulting ℱ is at most φ_n-1(0). Finally, we show that there do exist continuous functions g:ℝ^n→ℝ for which the order types of the resulting sets ℱ approach the small Veblen ordinal.
READ FULL TEXT
