{"title":"Herbrandized modified realizability","authors":"Gilda Ferreira, Paulo Firmino","doi":"10.1007/s00153-024-00917-6","DOIUrl":null,"url":null,"abstract":"<div><p>Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion of realizability, namely <i>herbrandized modified realizability</i>. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed <i>star combinatory calculus</i>, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"703 - 721"},"PeriodicalIF":0.3000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00917-6.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-024-00917-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 0
Abstract
Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion of realizability, namely herbrandized modified realizability. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic.
期刊介绍：
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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