0
votes
1answer
29 views

Type theoretic existential introduction and proof with subtypes

I'm working through a book[1], on type theory and categorial grammar (for linguistic applications). Sadly, I ran into problems pretty early on. I'd be very grateful if someone could Explain the ...
0
votes
0answers
33 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 ...
3
votes
2answers
100 views

Types versus kinds and sorts

In the context of logic, especially Higher‑Order‑Logic and Calculus‑of‑Construction, what is a kind and how does it relates to and differs from a type? My raw guess if that a kind is the higher level ...
3
votes
1answer
42 views

Basic Propositional Logic

I'm working through Thompson's Type Theory and Functional Programming. I've only read the first chapter and want to make sure I'm understanding the material. The first problem asks to prove the ...
1
vote
1answer
75 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 ...
3
votes
0answers
52 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 ...
8
votes
1answer
256 views

Axiom of Choice - Type Theory (Proof)

Background In Intuitionistic Type Theory (p. 27-28), Martin Löf provides a proof of the axiom of choice that is constructively valid. This version is considerably weaker than the ordinary set theory ...
2
votes
1answer
113 views

Unprovable Equivalence in Type Theory

Let $\prec$ be a binary relation on a set $A$… A predicate $P(x)$ set $(x:A)$ is said to be progressive with respect to $(A,\prec)$ if \begin{equation} (\forall a:A)\Big((\forall b:A)\big(b \prec a ...
5
votes
1answer
204 views

Foundational theories, their uses, interactions and comparisons?

Until now, I heard that there are some theories for building mathematical objects (or at least it is what it seems to my poor knowledge). Some of these are: Set theory; Logic; Category theory; Type ...
4
votes
1answer
119 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 ...
1
vote
1answer
60 views

How to construct a term of a particular type

I am reading the article "Introduction to Type Theory" by Herman Geuvers, the chapter explaining the Fitch style of natural inference. I stuck at the exercise 1.3 (first two are simple): ...
1
vote
1answer
98 views

What can be proven within Simply Typed lambda calculus?

I was reading http://en.wikipedia.org/wiki/Simply_typed_lambda_calculus and I'm having a hard time thinking of anything remotely interesting that can be proven within Simply Typed lambda calculus. Am ...
6
votes
1answer
754 views

Relationship between propositional logic, first-order logic, second-order logic higher-order logic, and type theory

I understand there is propositional logic, first-order logic, second-order logic higher-order logic, and type theory, where the latter logics are extensions of the former logics. Can someone explain ...
4
votes
2answers
155 views

Is higher order type theory the same as higher order logic?

The internal language of a topos is higher order intuitionistic type theory (or logic). Here the higher order simply refers to allowing function types. In mathematical logic we have higher-order ...
9
votes
0answers
170 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 ...
3
votes
3answers
227 views

Book on lambda calculus logic and type theory

Can someone recommend me a book for self study which will cover topics of logic, lambda calculus and type theory. I know about "Computability and Logic" written by Bolos but it describe recursive ...
5
votes
2answers
89 views

Is there a theory of extensible definitions?

We can define $+$ as a function $\mathbb{N}^2 \rightarrow \mathbb{N}$, and then prove: Theorem 1. The range of $+$ is $\mathbb{N}$. If we later wish to extend $+$ to a function $\mathbb{Z}^2 ...
2
votes
2answers
56 views

mapping simple first oreder problems to type theory

Since Peano axioms expressed in type theory doesn't seem to be going anywhere, here is a simpler question: How would I map simple first order systems to an equivalent type theoretic notation. ...
8
votes
1answer
424 views

Intuitionistic Banach-Tarski Paradox

While the Banach-Tarski paradox is a counter-intuitive result which requires the Axiom of Choice, leading some people to argue specifically against Choice, and others to argue for constructive ...
4
votes
2answers
758 views

Curry-Howard correspondence

I read that the Curry-Howard correspondence introduces an isomorphism between typed functions and logical statements. For example, supposedly the function $$\begin{array}{l} I : \forall a. a \to a\\ ...
12
votes
2answers
1k views

Classic type theory textbooks

There are many classic textbooks in set and category theory (as possible foundations of mathematics), among many others Jech's, Kunen's, and Awodey's. Are there comparable classic textbooks in ...
2
votes
2answers
247 views

Has a Dependent Type always a Type?

I am experimenting with dependent types. Lets assume the following short notation: ...
2
votes
0answers
79 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
2answers
316 views

formal rules for avoiding bound/unbound variable problems in lambda calculus

I have been interested in learning formal math formally enough so that I could write a proof assistant using some simple parsing tools, and explain to someone else with little math knowledge how to ...
43
votes
7answers
6k views

Learning Lambda Calculus

What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus? Specifically, I am interested in the following areas: Untyped lambda calculus ...