Models as Fundamental Entities in Set Theory: A Naturalistic and Practice-based Approach

This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins' claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy's naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to introduce models as new entities. In conclusion, I argue for a view that takes both sets and models to be fundamental entities in set theory.