Building on a result of Visser, Enayat identified a property of theories known as “solidity.” It was identified as a technical, model-theoretic property that implied the seemingly more natural property of “tightness”: roughly, there is no way to extend a tight theory in different ways that are saying essentially the same thing. Many foundational theories are both tight and solid; e.g., PA and ZFC. Proofs of tightness tend to start by establishing solidity first. Nonetheless, it has appeared (at least, to me) to be a merely technical property. 

In this talk, I want to do two quite different things. First, by appealing to the use of theory categories, I want to show that solidity is akin to a rigidity property for structures, but here the structures are categories of structures. I believe this makes solidity appear more intuitive and natural as a mathematical concept. Second, I want to question any simple analysis of its value in the foundations of mathematics. In particular, I will show that for set theories without full Separation and Replacement, solidity fails catastrophically. This is joint work with Elliot Glazer. Second, I’ll show that if forcing is admitted into the toolkit of interpretation (as set theorists tend to assume), then there is a natural way of seeing widespread failures of solidity for ZFC and its extensions.