A Model Theory for the Potential Infinite

We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely extensible finite. The main adoption is the interpretation of the u...

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Bibliographic Details
Published inReports on mathematical logic Vol. 57; no. 57; pp. 3 - 30
Main Author Eberl, Matthias
Format Journal Article
LanguageEnglish
Published Kraków Wydawnictwo Uniwersytetu Jagiellońskiego 01.01.2022
Jagiellonian University Press
Jagiellonian University-Jagiellonian University Press
Subjects
Dynamic models
Logic
Mathematics
Ontology
Philosophy of Science
first order logic
reflection principle
potential infinite
model theory
Finitism
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Summary:We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely extensible finite. The main adoption is the interpretation of the universal quantifier, which has an implicit reection principle. Each universal quantification refers to an indefinitely large, but finite set. The quantified sets may increase, so after a reference by quantification, a further reference typically uses a larger, still finite set. We present the concepts for classical first-order logic and show that these dynamic models are sound and complete with respect to the usual inference rules. Moreover, a finite set of formulas requires a finite part of the increasing model for a correct interpretation.
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ISSN:0137-2904
2084-2589
DOI:10.4467/20842589RM.22.001.16658