Lévy-Montague reflection is ${\Pi }_1^1$-conservative over ${\mathsf{WKL}}_0$

Abstract

We study a Lévy-Montague reflection scheme $\mathsf{Rfn}$ in second-order arithmetic: for each formula $\phi$, the scheme asserts that every set belongs to a countable coded $\omega$-model such that $\phi$ is absolute, at all parameters from the model, between the model and the universe. Our central result is a model extension construction: every countable model of ${\mathsf{RCA}}_0$ can be extended, without changing its first-order part, to a model of ${\mathsf{WKL}}_0$ together with the full scheme $\mathsf{Rfn}$. It follows at once that ${\mathsf{WKL}}_0+\mathsf{Rfn}$ is ${\Pi }_1^1$-conservative over both ${\mathsf{WKL}}_0$ and ${\mathsf{RCA}}_0$, that its first-order part is exactly $\mathrm{I}{\Sigma }_1$, and that it is ${\Pi }_2^0$-conservative over $\mathsf{PRA}$. The result opens an avenue for adopting, within a theory conservative over $\mathsf{PRA}$, Feferman's $\mathsf{ZFC}$-formalization of universe-based category-theoretic arguments that was achieved using Lévy-Montague reflection. The conservation proof itself, however, is non-finitary. The extension is the union of an ${\omega }_1$-tower of forcing extensions, and its uncountable cofinality is what secures reflection. We are only able to prove the conservation in $\mathsf{PRA}+\text{1-Con}({\mathsf{Z}}_2)$. The results were obtained with extensive use of Anthropic's large language model Fable 5.

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