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 $\beta$-models: $T\prec_\beta U$ if every $\beta$-model of $U$ contains a countable coded $\beta$-model of $T$. The restriction of $\prec_\beta$ to theories with $\beta$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_\beta$-rank. Second, the $\prec_\beta$-ranks of $\mathcal{L}_\in$ theories are cofinal in $\omega_1$. Third, assuming $V=L$, the $\prec_\beta$-ranks of $\mathcal{L}_2$ theories are cofinal in $\omega_1$. Finally, $\delta^1_2$ is the supremum of the $\prec_\beta$-ranks of finitely axiomatized theories.
From: James Walsh [view email]
[v1]
Wed, 26 Mar 2025 11:52:19 UTC (25 KB)
[v2]
Wed, 24 Jun 2026 17:59:47 UTC (17 KB)
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