TY - JOUR
TI - A Model Theory for the Potential Infinite
AU - Eberl, Matthias
TI - A Model Theory for the Potential Infinite
AB - 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.
VL - 2022
IS - Number 57
PY - 2022
SN - 0137-2904
C1 - 2084-2589
SP - 3
EP - 30
DO - 10.4467/20842589RM.22.001.16658
UR - https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/a-model-theory-for-the-potential-infinite
KW - Finitism
KW - potential infinite
KW - model theory
KW - first order logic
KW - reflection principle