Ranking theories via encoded $β$-models

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.