While both Elixir and TypeScript are structural, they are different type systems. A set-theoretic type system is not one that has unions, intersections, negations, but rather one where the foundation of the type system is represented on top of unions, intersections, and negations. Even relations such as subtyping, compatibility, etc. are expressed as set-theoretic. We also have a very different approach to dynamic/gradual typing (which is sound): https://elixir-lang.org/blog/2023/09/20/strong-arrows-gradua...
I recommend reading Giuseppe Castagna's work for those who want to dig deeper: https://www.irif.fr/~gc/
I did read your article, and it’s actually how I came out with the idea that Elixir had come up with the same idea as TS haha!
I’m sure I’m missing something as I have no doubt you’re by far more knowledgeable than I am, but TS also represents types as sets (semantically, dunno about internal implementation), structural typing seems to inherently map to set theory
For example, about bounded polymorphism the article says “Of course, we can provide syntax sugar for those constraints”: seems like this sugar could be ‘a extends number’ to me, like typescript does. I don’t see how TS does _not_ map to set operations
Sorry for the late reply but I wanted to double check before following up. And I am glad I did because I was told my first response was inaccurate. :)
It is fair to call both TypeScript and Elixir as set-theoretic type system. The big difference is that Elixir's implements semantic subtyping. Here is the paper you can use to dig deeper. I hope it helps: https://www.irif.fr/~gc/papers/lics02.pdf
I recommend reading Giuseppe Castagna's work for those who want to dig deeper: https://www.irif.fr/~gc/