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As @cody already pointed out, you cannot hope to have such a surjection inside type theory because nat → nat is always internally uncountable by Cantor's diagonalization, i.e., given any alleged surje …

I would recommend looking at higher-order logics (HOL), for instance Lambek and Scott's Introduction to Higher-Order Categorical Logic. Such a logic will set up binding of bound variables and treat va …

Computation is about information processing. The intrinsic nature of information and information processing naturally leads to topological notions (see Neel's answer about domain theory), but these ar …

I would like to supplement the answer by cody by a general observation conveying my understanding of why the type checking algorithms work. For a wide class of type theories, type checking or inferen …

While I am not aware of any such results, other than your own, I think you could broaden the scope somewhat and ask: what results in TCS have been proved using (any kind of) non-standard axioms. By no …

I recommend that you look at realizability theory. In realizability computational models are known as partial combinatory algebras (PCA). They cover a wide range of computational models. There is a 2- …

Cantor space is used quite widely in the theory of representations of topological spaces in general, and not just the real numbers. An important realizability model, namely Kleene's function realizab …

Is chapter 29 of Bob Harper's book what you are looking for?

What you are asking for does not exist for a general-purpose programming language (by which we mean that the language can simulate Turing machines, and that Turing machines can simulate the language). …

I am not an expert on this at all, but I would like to get a constructive discussion started. Here is an attempt, based on the math.stackexchange.com question Count the number of positive solutions fo …

No, it is not decidable. A good heuristic to answer such questions is the following: every computable map is continuous. If you could decide whether $f(x) = 0$ for all $x \in \mathbb{C}$, then the cha …

"We use a suitable Gödel numbering of descriptions of context-sensitive grammars. For example, a context-sensitive grammar may be represented by a string of characters in some accepted formalism. Obvi …

The first theorem of the form you are asking about was proved by Y. Moschovakis in Notation systems and recursive ordered fields, Compositio Mathematica 17:40–71 (1965). Then in the context of Type Tw …

The axiom of choice is used when there is a collection of "things" and you choose one element for each "thing". If there is just one thing in the collection, that's not the axiom of choice. In our cas …

Perhaps Dale Miller's Overview of linear logic programming is a good strarting point?