For questions about type theory, including normalization, dependent types, identity types, inductive types, universe types, functional programming languages, proofs as programs in simply-typed lambda-calculi, Martin-Löf's intuitionistic type theory or related.Consider using one of the following ...

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13
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1answer
358 views

Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.

Is there a rigorous approach to mathematical notation wherein the "universe" is divided into disjoint "worlds," and the meaning of notation is world-dependent? This would solve a few pesky problems. ...
4
votes
1answer
140 views

Curry-Howard Correspondence (Proof Theory)

As you all know, the Curry-Howard correspondance provides a link between type theory and predicate logic. Concepts featured in the former, such as $\Pi$-type and $\Sigma$-type can, by the ...
2
votes
1answer
80 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
1
vote
1answer
70 views

Mistake in http://plato.stanford.edu/entries/type-theory/#2

There seems to be mistake in http://plato.stanford.edu/entries/type-theory/#2: First-order logic considers only types of the form $i,…,i → i$ (type of function symbols), and $i,…,i → o$ ...
0
votes
1answer
93 views

Define inl : $σ → σ ∨ τ$

I'm a bit stuck in Geuvers' "Introduction to Type Theory" (http://www.cs.ru.nl/~herman/onderwijs/provingwithCA/paper-lncs.pdf), p. 39: Exercise 13. Prove the derivability of some of the other logical ...
11
votes
0answers
194 views

Reference on standard types

This question is about what I presume is a basic construction in type theory. The finite types are defined as follows: 0 is a finite type; if $\sigma, \tau$ are finite types, then so is ...
6
votes
0answers
50 views

What is the actual significance of the lambda calculus for the formalization of math?

The Simply Typed Lambda Calculus was proposed initially as a foundational system for the formalization of mathematics. As such, I would expect that soon there would be attempts to implement most of ...
5
votes
0answers
63 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
5
votes
0answers
61 views

Path structure of an odd-looking higher inductive type.

Consider the (putative) higher inductive type $\mathtt{PNest}$, with the constructors: $\mathtt{base} : \mathtt{PNest}$ $\mathtt{nest} : \mathtt{base} = \mathtt{base} \to \mathtt{base} = ...
4
votes
0answers
97 views

Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
4
votes
0answers
111 views

W-types and inverse image functor

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved. I would like to know whether the ...
3
votes
0answers
122 views

What is the likely future of Univalent Foundations?

Univalent foundations has been hyped up as the foundation for mathematics for the future in articles such as this one. Now I've given HoTT a brief look, and at least seen that it appears on the face ...
3
votes
0answers
70 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
3
votes
0answers
95 views

Are large cardinals bi-interperable with type theory?

"Sets in Types, Types in Sets" establishes that the calculus of constructions is bi-interpretable with ZFC + infinitely many inaccessibles. Are there any more results like this higher up the large ...
2
votes
0answers
32 views

Proof by reflection and Homotopy Type Theory

I have been looking into various proof assistants, and came across the method of proof by reflection in Coq, which (from what I understand) allows one to verify a program provides the correct "answer" ...
2
votes
0answers
31 views

What strongly normalizing lambda calculi exist that can be integrated with/as logic?

If I'm trying to implement a logical system for deduction based on propositional reasoning, I can start with predicates and quantifiers and functions to obtain first order logic. I can further extend ...
2
votes
0answers
42 views

When is the higher-order theory of a model categorical?

I'm interested in (classical) type theories $L$ with the following property. For $M$ any model of $L$ (in Set), let $T(M)$ be the type theory of $M$, i.e., the strongest extension of $L$ satisfied by ...
2
votes
0answers
121 views

Continuations vs. Yoneda

There is this observation (worldpress blog) that the continuation monad (Wikipedia) stems from the Yoneda embedding. For fixed data types $R$, the monad maps data types $T$ to types $(T\to R)\to R$. ...
2
votes
0answers
44 views

What if typing judgements were propositions?

Suppose we created a modified type theory in which typing judgments like $x : X$ are not judgments at all; rather, they're propositions. Would this be a bad idea? If so, why? A good answer should ...
2
votes
0answers
66 views

Set Theory - Well Order (Lexiographical combination)

Question: Prove constructively that if $(A_{1},\prec_{1})$ and $(A_{2},\prec_{2})$ are two well-ordered sets then their lexicographical combination $(A_{1} \times A_{2},<_{1,2})$ is also well ...
2
votes
0answers
81 views

What are the main differences between set theory versus pure type systems?

According to wikipedia, a pure type system: ...is a form of typed lambda calculus that allows an arbitrary number of sorts and dependencies between any of these... Pure type systems may ...
2
votes
0answers
90 views

Introductory text about different stratification methods in higher-order logic and set theory

Could someone recommend me a good overview text about stratification of predicates, comprehension axioms, and other methods of avoiding the paradoxes in untyped or only loosely/relatively typed ...
1
vote
0answers
25 views

Proving the Identity Type Has a Term

I am trying to get used to type theory. I have a two things to ask. Let $1$ be the unit type with unique term $\star : 1$. As an exercise I was trying to prove that $$ ...
1
vote
0answers
55 views

Proof of $(\forall x. \varepsilon(x)) \Rightarrow \bot $ in $\lambda\pi $ calculus $\equiv$

What is the right representation of the proof of $(\forall x. \varepsilon(x)) \Rightarrow \bot $ in simple type theory as a term of $\lambda\pi $ calculus $\equiv$? Note on notation: The epsilon ...
1
vote
0answers
42 views

Reductions under lambda in dependently typed lambda calculus

I am currently reading a Simon Thompson's Type Theory book. In chapter 5 he introduces a system TT(0,C), which limits a notion of reduction . In this notion of reduction, reductions under lambda are ...
1
vote
0answers
50 views

Enumeration of symbols in grammatical expressions or vertices in tree graphs

I have expressions (type of a function) like e.g. $$f:(A\to B)\to C \to (D\to E)\to F.$$ (Where I understand $A\to B\to C$ as $A\to (B\to C)$, in case that is relevant.) There might be information ...
0
votes
0answers
15 views

Structural induction over types that accept functions, in Coq

If you define an inductive type in Coq with a constructor that accepts a function mapping to that type, you get a somewhat odd induction rule. ...
0
votes
0answers
31 views

Proper terminology for a pair of magmas

Given two magmas, call them $f$ and $g$ over the same set $a$, what do I call a tuple $h$ of them? With extra properties we call them such names as fields and rings. I am just looking for the generic ...
0
votes
0answers
37 views

Using types instead for basic proofs

Disclaimer: I know close to nothing about formal type theory, but I program intensively so prefer to think in terms of types. In typical math you write $a \in A$ to speak of the nature of your ...