This comment contains a lot of false information. I’m first going to point out that there is a model of Lean’s type theory called the cardinality model, in which all types of equal cardinality are modelled as the same set. This is why I say the types have no useful distinguishing characteristics: it is consistent to add the axiom `Nine = Unit` to the type theory. For the same reason, it is consistent to add `ℕ = ℤ` as an axiom.

> So a Nine type with a single constructor is indeed isomorphic to Unit but it's not the same type, it carries different syntactic and semantic meaning and the type system preserves that.

It carries different syntax but the semantics are the exact same.

> Type theory is usually intensional not extensional so two types with the same number of inhabitants can have wildly different structures

It is true that type theory is usually intensional. It is also true that two types equal in cardinality can be constructed in multiple different ways, but this has nothing to do with intensionality verses extensionality – I wouldn’t even know how to explain why because it is just a category error – and furthermore just because they are constructed differently does not mean the types are actually different (because of the cardinality model).

> Cardinality is a set-theoretic notion but most type theories are constructive and syntactic, not purely set-theoretic.

I don’t know what you mean by this. Set theory can be constructive just as well as type theory can, and cardinality is a fully constructive notion. Set theory doesn’t have syntax per se but that’s just because the syntax is part of logic, which set theory is built on.

> most type theories don't enforce full parametricity at runtime

What is “most”? As far as I know Lean does, Coq does, and Agda does. So what else is there? Haskell isn’t a dependent type theory, so it’s irrelevant here.

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Geniune question: Where are you sourcing your information from about type theory? Is it coming from an LLM or similar? Because I have not seen this much confusion and word salad condensed into a single comment before.