A Model Theory for the Potential Infinite

Matthias Eberl

Abstrakt

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.

AMS subject classification: 03C68, 03C30, 03C13.

Słowa kluczowe: Finitism, potential infinite, model theory, first order logic, reflection principle
References

[1] T.J. Carlson, Ranked partial structures, The Journal of Symbolic Logic 68:4 (2003), 1109-1144.

[2] M. Dummett, What is Mathematics About?, in: Mathematics and Mind, A. George (Ed.), Oxford University Press 1994, pp. 11-26.

[3] M. Eberl, Infinity is not a Size, in: The Logica Yearbook 2020, I. Sedlfiar and M. Blicha (Eds.), College Publications 2021, pp. 33-48

[4] S. Lavine, Understanding the infinite, Harvard University Press 2009

[5] O. Linnebo and S. Shapiro, Actual and potential infinity, Noûs 53:1 (2019), 160-191.

[6] M. Mostowski, On representing semantics in finite models, in: Philosophical dimensions of logic and science, A. Rojszczak, J. Cachro, G. Kurczewski (Eds.), Springer 2003, pp. 15-28.

[7] M. Mostowski, Truth in the limit, Reports on Mathematical Logic 51 (2016), 75-89.

[8] J. Mycielski, Locally Finite Theories, Journal of Symbolic Logic 51:1 (1986), 59-62.

[9] J. Mycielski, The meaning of pure mathematics, Journal of Philosophical Logic 18:3 (1989), 315-320.

[10] K-G. Niebergall, Assumptions of infinity, in: Formalism and beyond: on the nature of mathematical discourse, G. Link (Ed.), De Gruyter 2014, pp. 229-274.

[11] H. Putnam, Mathematics without foundations, The Journal of Philosophy 64:1 (1967) 5-22.

[12] S. Schapiro and C. Wright, All things indefinitely extensible, in: Absolute generality, A. Rayo and G. Uzquiano (Eds.), Oxford University Press 2006, pp. 255-304.

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.