Categoricity in V=L for recursive complete second‑order theories

Determine whether, assuming V = L, every recursively axiomatized complete second‑order theory is categorical; that is, establish whether all models of any recursively axiomatized complete second‑order theory are isomorphic in the constructible universe.

Background

The paper investigates the categoricity of complete second‑order theories under various set‑theoretic hypotheses. Under V = L, classical results imply strong categoricity phenomena for finitely axiomatized theories (e.g., Solovay’s result that such theories are categorical) and for countable structures, while adding reals can yield non‑categorical behavior.

The authors provide numerous positive and negative results under assumptions such as PD, AD, (∗), and L[U], but the status in L for recursively axiomatized complete second‑order theories remains unresolved. This question, posed by Solovay, asks whether the constructible universe enforces categoricity even for recursively axiomatized complete theories.