Ordinals and recursively defined functions on the reals

Abstract

We determine sufficient conditions under which certain recursively defined functions are well defined for all real inputs. Given a function $f:\mathbb R\to\mathbb R$, call a decreasing sequence $x_1>x_2>x_3>\cdots$ "$f$-bad" if $f(x_1)>f(x_2)>f(x_3)>\cdots$, and call the function $f$ "ordinal decreasing" if there exist no infinite $f$-bad sequences. We prove the following result: Given ordinal decreasing functions $f,g_1,\ldots,g_k,s$ that are everywhere larger than $0$, define the recursive algorithm "$M(x)$: if $x<0$ return $f(x)$, else return $g_1(-M(x-g_2(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$". Then $M(x)$ halts and is ordinal decreasing for all $x \in \mathbb{R}$. The recursive algorithms $M$ and $M_n$ previously studied in the context of fusible numbers by Ericskon et al. (2022) and Bufetov et al. (2024), respectively, are special cases of this scheme. Moreover, given an ordinal decreasing function $f$, denote by $o(f)$ the ordinal height of the root of the tree of $f$-bad sequences. Then we prove that, for $k\ge 2$, the function $M(x)$ defined by the above algorithm satisfies $o(M)\le\varphi_{k-1}(\gamma+o(s)+1)$, where $\gamma$ is the smallest ordinal such that $\max\{o(s),o(f),o(g_1), \ldots, o(g_k)\} <\varphi_{k-1}(\gamma)$.