Exploring Epstein's Mathematical World: Models, Logic, and Infinity

Let's dive into the curious world of Epstein semantics, a fascinating topic in mathematics. One of the key tools used in this exploration is the model-theoretic S-set construction, introduced by Krawczyk in 2022. This tool helps us uncover several interesting facts. First, it shows that there are countless Epstein models that behave exactly the same way. This means that for every Epstein model, there are an infinite number of models that are indistinguishable from it. Next, we find that the logic governing these generalized Epstein models is a special fragment of classical propositional logic (CPL). This is similar to a famous result in modal logic by Johan van Benthem. But things get even more intriguing! Some sets of Epstein relations can't be defined, no matter how hard we try. Yet, the logics describing these undefinable sets can be neatly summarized using just a few rules. The story doesn't end there. We also discover that there are countless Epstein-incomplete logics, meaning they can't express all true statements. Additionally, the logic of generalized Epstein models has a cool property called interpolation, which helps us deduce certain statements from others.