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Comment by sa1 | Hacker News Reader

What do set-theoretic types mean? Aren’t types an alternative approach meant to avoid the paradoxes with sets?

Is it just being used as a marketing term?

Short answer: “a type system centered on the use of set-theoretic types (unions, intersections, negations) that satisfy the commutativity and distributivity properties of the corresponding set-theoretic operations”.

Long answer, well, there are blog posts[0], the Design Principles of the Elixir Type System paper[1] and related presentations[2, 3, 4] that talk about it at length. Giuseppe Castagna’s site has many more related papers: https://www.irif.fr/~gc/topics.en.html

[0]: https://elixir-lang.org/blog/2022/10/05/my-future-with-elixi...

[1]: https://www.irif.fr/~gc/papers/elixir-type-design.pdf

[2]: https://www.youtube.com/watch?v=gJJH7a2J9O8

[3]: https://www.youtube.com/watch?v=VYmo867YF6g

[4]: https://www.youtube.com/watch?v=giYbq4HmfGA

Sets and types are foundational mathematical concepts so I’m looking for how elixir’s types fit in that context. Union and intersection are not something that belongs only to sets.

It means that the types are built on unions, intersections, and negations[1]. It's a polymorphic type system with inference at the function level. It also does some type narrowing with pattern matching.

[1] https://www.irif.fr/_media/users/gduboc/elixir-types.pdf

Unions, intersections and negations are available in types as well and are by no means exclusive to sets. The distinguishing feature of a set vs type is that a value belongs to just one type while it can belong to several sets.

Types do not inherently have any such restrictions. A value can belong to several types. In fact, if you posit types to have union, that necessarily follows.

I think they do, and as you mentioned you can explicitly remove such a restriction. Sets and types are once again two different kinds of objects in mathematical theory, and a set-theoretic type doesn’t seem to be based either on set theory or type theory.

If types have unions/intersections/negations, as you originally seemed to imply, then “a value belongs to just one type” is false (if x is of type A then it’s also of type A ∪ B for any B).

If they don’t unless you add them to “explicitly remove such a restriction”, then that means you’re making types more set-like (set-theoretic).

In strict “type theory”, it’s the latter: types don’t have unions (in the sense that set-theoretic types do). There are sum types, which are different.

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