Ranking theories via encoded $β$-models
Hanul Jeon, Patrick Lutz, Fedor Pakhomov, James Walsh
TL;DR
The paper introduces a β-model–based semantic ranking of theories, defining $T \prec_β U$ to mean every $β$-model of $U$ encodes a $β$-model of $T$, and proves this ordering is well-founded with countable ranks. It establishes that each theory has a countable β-rank, shows cofinality results for $L_\in$ theories in $\omega_1$ and, under $V=L$, for $L_2$ theories as well, and identifies $\delta^1_2$ as the supremum of ranks for finitely axiomatized theories; it further computes β-ranks for several key theories. The work connects model-theoretic strength with set-theoretic tools such as admissibles and constructibility, yielding a robust semantic hierarchy that complements traditional proof-theoretic and reverse-m mathematics perspectives. These results illuminate how encoded β-models capture proof-theoretic strength in a well-founded, ordinal-indexed framework with implications for the study of subsystems of second-order arithmetic and set theory.
Abstract
Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $β$-models: $T\prec_βU$ if every $β$-model of $U$ contains a countable coded $β$-model of $T$. The restriction of $\prec_β$ to theories with $β$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_β$-rank. Second, the $\prec_β$-ranks of $\mathcal{L}_\in$ theories are cofinal in $ω_1$. Third, assuming $V=L$, the $\prec_β$-ranks of $\mathcal{L}_2$ theories are cofinal in $ω_1$. Finally, $δ^1_2$ is the supremum of the $\prec_β$-ranks of finitely axiomatized theories.