FROM REAL ANALYSIS TO THE SORITES PARADOX VIA REVERSE MATHEMATICS
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| Názov: | FROM REAL ANALYSIS TO THE SORITES PARADOX VIA REVERSE MATHEMATICS |
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| Autori: | WALTER DEAN, SAM SANDERS |
| Zdroj: | The Review of Symbolic Logic. :1-27 |
| Publication Status: | Preprint |
| Informácie o vydavateľovi: | Cambridge University Press (CUP), 2025. |
| Rok vydania: | 2025 |
| Predmety: | Mathematics - History and Overview, History and Overview (math.HO), FOS: Mathematics, Mathematics - Logic, Logic (math.LO) |
| Popis: | This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional discrete formulations are reliant on Hölder’s representation theorem for ordered Archimedean groups. While this is provable in $\mathsf {RCA}_0$ , we also consider two forms of the sorites which rest on non-constructive principles: the continuous sorites of Weber & Colyvan [35] and a variant we refer to as the covering sorites. We show in the setting of second-order arithmetic that the former depends on the existence of suprema and thus on arithmetical comprehension ( $\mathsf {ACA}_0$ ) while the latter depends on the Heine–Borel theorem and thus on Weak König’s lemma ( $\mathsf {WKL}_0$ ). We finally illustrate how recursive counterexamples to these principles provide resolutions to the corresponding paradoxes which can be contrasted with supervaluationist, epistemicist, and constructivist approaches. |
| Druh dokumentu: | Article |
| Jazyk: | English |
| ISSN: | 1755-0211 1755-0203 |
| DOI: | 10.1017/s1755020325000061 |
| DOI: | 10.48550/arxiv.2502.06596 |
| Prístupová URL adresa: | http://arxiv.org/abs/2502.06596 |
| Rights: | Cambridge Core User Agreement arXiv Non-Exclusive Distribution |
| Prístupové číslo: | edsair.doi.dedup.....7079db0ceba520f89b22cc62e0c44d38 |
| Databáza: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional discrete formulations are reliant on Hölder’s representation theorem for ordered Archimedean groups. While this is provable in $\mathsf {RCA}_0$ , we also consider two forms of the sorites which rest on non-constructive principles: the continuous sorites of Weber & Colyvan [35] and a variant we refer to as the covering sorites. We show in the setting of second-order arithmetic that the former depends on the existence of suprema and thus on arithmetical comprehension ( $\mathsf {ACA}_0$ ) while the latter depends on the Heine–Borel theorem and thus on Weak König’s lemma ( $\mathsf {WKL}_0$ ). We finally illustrate how recursive counterexamples to these principles provide resolutions to the corresponding paradoxes which can be contrasted with supervaluationist, epistemicist, and constructivist approaches. |
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| ISSN: | 17550211 17550203 |
| DOI: | 10.1017/s1755020325000061 |