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I'm aware of strongly normalizing systems and the escape hatch of coinductive programming. But when we're talking about the space of all programs, the fundament

I'm aware of strongly normalizing systems and the escape hatch of coinductive programming. But when we're talking about the space of all programs, the fundamental limit of incompleteness is important. How else do we judge the merit of a type system except by seeing how it fits into the overall space of computable processes? There are two ways to see type systems. In the first way you construct terms along with their types, this is called Church style. In the second way, the terms exist before their types and you use types to describe their behavior, this is called Curry style. In particular take System F. In Church style the terms of System F come with their types. In Curry style we see System F types as a way to describe the behavior of untyped lambda terms. I used to think Church style was more important but lately I've been more partial to Curry style. Programs exist before you type them, type systems tell you how they behave. They also tell you how to construct programs but this is subordinate to the more fundamental descriptive capacity.

tom_mellior 9y ago
> But when we're talking about the space of all programs, the fundamental limit of incompleteness is important. I agree with this. But I think that usually we are not really talking about all programs. We are talking about useful programs, and those are usually terminating. In theory, not always, see first-order theorem provers; but in practice, we always call a prover with a time limit because nontermination isn't useful. > Programs exist before you type them, type systems tell you how they behave. They also tell you how to construct programs but this is subordinate to the more fundamental descriptive capacity. That's an interesting point, and I agree in many respects. I think when programming in a dynamically typed language, I approach things in one style, and in statically typed languages in another style. But specifically for termination, I don't think so. I never want to write a nonterminating program; the termination property for my programs exists (as a requirement) before the program does.

tom_mellior 9y ago

> But when we're talking about the space of all programs, the fundamental limit of incompleteness is important. I agree with this. But I think that usually we are not really talking about all programs. We are talking about useful programs, and those are usually terminating. In theory, not always, see first-order theorem provers; but in practice, we always call a prover with a time limit because nontermination isn't useful. > Programs exist before you type them, type systems tell you how they behave. They also tell you how to construct programs but this is subordinate to the more fundamental descriptive capacity. That's an interesting point, and I agree in many respects. I think when programming in a dynamically typed language, I approach things in one style, and in statically typed languages in another style. But specifically for termination, I don't think so. I never want to write a nonterminating program; the termination property for my programs exists (as a requirement) before the program does.