This is a talk in the New England Recursion and Definability Seminar on 2024 November 17.

[slides]

Every logician knows that the Löwenheim–Skolem theorem rules out the possibility of absolute categoricity result for theories formulated in first-order logic. This is in contrast to the Dedekind and Zermelo theorems about the categoricity of arithmetic and set theory in second-order logic. Nevertheless, when something is impossible we still want to see how close we can get. The titular three properties, originating in work by Visser and named by Enayat, are notions of semantic completeness. Informally speaking, they capture the idea that a theory cannot be extended in the same signature to have new semantic content. Thus they form a categoricity-like property which is enjoyed by important foundational theories such as Peano arithmetic.

In this talk I will survey known results in this area, both positive and negative. A key empirical fact is that these categoricity-like properties seem to characterize canonical theories like PA or ZF. I will focus the examples for the talk subsystems of second-order arithmetic.