{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-escape-from-vardanyan-s-theorem","title":"An Escape from Vardanyan's Theorem","arxiv_id":"2102.13091","date":"2021-02-25","proceeding":null,"authors":["Ana de Almeida Borges","Joost J. Joosten"],"abstract":"Vardanyan's Theorems state that $\\mathsf{QPL}(\\mathsf{PA})$ - the quantified provability logic of Peano Arithmetic - is $\\Pi^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\\mathsf{QRC}_1$ was previously introduced by the authors as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\\mathsf{QRC}_1$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\\mathsf{QRC}_1$ is the strictly positive fragment of $\\mathsf{QGL}$ and a fragment of $\\mathsf{QPL}(\\mathsf{PA})$.","url_abs":"https://arxiv.org/abs/2102.13091v3","url_pdf":"https://arxiv.org/pdf/2102.13091v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-escape-from-vardanyan-s-theorem","repo_url":"https://gitlab.com/ana-borges/QRC1-Coq","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}