Ordinals and recursively defined functions on the reals (2311.17210v3)

Abstract: 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, which generalizes results of Erickson et al. (2022) and Bufetov et al. (2024): 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 $x \in \mathbb{R}$. More specifically, 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 $\max{{o(s),o(f),o(g_1), \ldots, o(g_k)}}<\varphi_{k-1}(\gamma)$.