{"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/well-orders-in-the-transfinite-japaridze","title":"Well-orders in the transfinite Japaridze algebra","arxiv_id":"1212.3468","date":"2012-12-14","proceeding":null,"authors":["David Fernández-Duque","Joost J. Joosten"],"abstract":"This paper studies the transfinite propositional provability logics $\\glp_\\Lambda$ and their corresponding algebras. These logics have for each ordinal $\\xi< \\Lambda$ a modality $\\la \\alpha \\ra$. We will focus on the closed fragment of $\\glp_\\Lambda$ (i.e., where no propositional variables occur) and \\emph{worms} therein. Worms are iterated consistency expressions of the form $\\la \\xi_n\\ra \\ldots \\la \\xi_1 \\ra \\top$. Beklemishev has defined well-orderings $<_\\xi$ on worms whose modalities are all at least $\\xi$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original $<_\\xi$ orderings and provide a calculus for the corresponding generalized order-types $o_\\xi$. Our calculus is based on so-called {\\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form $\\la {\\formerOmega}_\\xi (A) \\ra_{\\xi \\in \\ord}$ for some worm $A$. One of these characterizations is in terms of a second kind of transfinite iteration called {\\em cohyperation.}","url_abs":"https://arxiv.org/abs/1212.3468v4","url_pdf":"https://arxiv.org/pdf/1212.3468v4.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":"well-orders-in-the-transfinite-japaridze","repo_url":"https://gitlab.com/ana-borges/wormscoq","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}