2025/03/26 by Jeon, Hanul, Lutz, Patrick, Pakhomov, Fedor +1
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2503.20470
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\precβU if every β-model of U contains a countable coded β-model of T. The restriction of \precβ 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 \precβ-rank. Second, the \precβ-ranks of L_∈ theories are cofinal in ω1. Third, assuming V=L, the \precβ-ranks of L2 theories are cofinal in ω1. Finally, δ12 is the supremum of the \precβ-ranks of finitely axiomatized theories.