[Kutta]

Incorrect. Whether the language as a logic is capable of expressing undecidable statements, is orthogonal to whether type checking is decidable. Type checking is analogous to checking proof validity in logics. Commonly used proof systems for first-order or higher-orders logics admit decidable proof validity, and already many first-order theories allow expressing undecidable statements. ZFC is a rather obvious example. Likewise intensional type theory is highly expressive as a logic and admits decidable proof validity.

Decidability of type checking is instead related to how explicit surface syntax is. In intensional type theory, the usage of equality proofs is marked explicitly, which is sufficient to retain decidable type checking. In contrast, in extensional type theory, the usage of equality proofs is left implicit, which causes type checking to be undecidable.

[johncolanduoni]

This is not true, as MLTT-style type theories have a different syntactical notion of proof than first order logical theories like ZFC. Proof of termination of type checking for Idris is equivalent to proof that all terms can be reduced to a normal form: for any given term, you can construct another term which when type checked requires fully normalizing the first term. It is well known that proving that the normal form evaluation of all terms in a MLTT type theory terminates is equivalent to the system’s consistency (because the false type has no constructors). This is how proofs of relative consistency of these type theories are usually proved.

[Kutta]

I don't see that "normalization implies consistency", which I'm aware of, relates to my previous comment in any relevant way.

ZFC and MLTT do not differ in that decidability of proof validity is not related to logical expressiveness.

It's not even true that for type theories, decidability of proof validity implies normalization. Normalization is not logically equivalent to decidable type checking.

For example, we can have a term language for extensional type theory which is annotated with reduction traces. This is the kind of syntax that we get when we embed ETT inside ITT. It has decidable type checking, as the type checker does not have to perform reduction, it only has to follow traces. This kind of tracing is actually used in the GHC Haskell compiler to some extent. So we have decidable type checking for a surface syntax of a type theory, for a type theory which is a) consistent b) not strongly normalizing.

[bmc7505]

I think you might be confusing the language and the type system. Not saying anything about the runtime characteristics of the language (which is obviously Turing Complete), but the type system itself, which is TC. If the type system allows you to encode a TM, then it is undecidable, regardless of whether it terminates due to an incomplete type checker. Since the Idris type system allows you to encode a TM, then the type system is undecidable due to the Halting problem.

[Kutta]

A type system cannot be TC. What you seem to talk about, is that if a type checker can simulate arbitrary TM's through an encoding of its input, then the type checker is necessarily non-total. This is correct. But the Idris checker is total and it is not possible to use it to simulate TMs.

The Idris type system does allow you to specify and formalize TMs internally, but that has no bearing on decidability of type checking, as I explained before.