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0
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0answers
26 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 ...
2
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1answer
50 views

Is it possible to express sigma-type in Martin-Löf type theory with other constructs

In Martin-Löf type theory we have sigma types (dependent products). Is it possible to express them with other constructs? How expressive is dependently typed lambda calculus without them, i.e. what we ...
1
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0answers
38 views

Is the (type)class `Functor` itself a functor? [closed]

Simple yes or no question, but one that's hard to google/search for due to repetition of terms: Since Functor is the set (ok, class) of all types for which an ...
9
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1answer
242 views

Type theory as foundations

Does anyone know any good references that describe type theoretical foundations of mathematics? I've read some books e.g. Winskel's The Formal Semantics of Programming Languages and Pierce's Types and ...
7
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4answers
161 views

How or why does intutionistic logic proof negations from within the theory, constructively?

I'm having a little of a cognitive dissonance why, in intuitionistic logic, it's possible to show stentences like $(\neg A \land \neg B) \implies \neg(A\lor B).$ In plain text: If 'A isn't true' as ...
5
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1answer
120 views

How to prove that these are the only type inhabitants?

Consider a type $$\mathsf{Boolean} = \forall \alpha.\ \alpha \to \alpha \to \alpha$$ with its two inhabitants \begin{align} \mathrm{tt} &= \lambda x.\ \lambda y.\ x \\ \mathrm{ff} &= \lambda ...
8
votes
1answer
255 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. ...
2
votes
2answers
98 views

Variables in Types in type theory

I'm slowly grasping this, though the different formulations of type theory make it difficult. In http://imps.mcmaster.ca/doc/seven-virtues.pdf types can only be formed from *, i, and a->b when a and ...
2
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1answer
62 views

How do definitions work in Martin-Lof type theory?

The classical viewpoint is that we can found mathematics by specifying a formal system $F$ whose theorems are precisely those of ZFC. However, since $F$ has essentially no support for the concept of a ...
5
votes
1answer
194 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 ...
5
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3answers
87 views

Types, Sets and Categories

I am learning Category Theory and at first I just pictured a category like a class in object oriented programming: type definition + methods (morphisms). However the author I am reading uses maps ...
1
vote
1answer
57 views

Typed Category Theory?

This small book (or long paper) describes objects and morphisims in a cateogry as having types. He also talks about pre-categories as categories with typeless objects and morphisims. While I am ...
3
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2answers
93 views

Is there a (foundational) type theory with the features I'm looking for?

I like to distinguish between sets and subsets. We imagine that sets are floating free in the universe, and that the elements of a set are constructed according to some kind of recursive rules. Like ...
0
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0answers
29 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 ...
3
votes
1answer
38 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 ...
9
votes
1answer
244 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
79 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
1
vote
1answer
74 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 ...
2
votes
2answers
609 views

Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category

I've been trying to find a proof that the pullback functors in a locally cartesian closed category have right adjoints (used to model the notion of indexed product inside a category (rather than ...
4
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0answers
102 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
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0answers
46 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 ...
2
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0answers
37 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
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1answer
67 views

Simple type theory: Proof inexistance of closed term

In simple type theory, how can I prove that there is no closed term of type? $$((P \Rightarrow Q) \Rightarrow Q) \Rightarrow P$$
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0answers
42 views

Find subgraphs in a directed graph which are isolated by edge properties

Please excuse my small knowledge of graph theory vocabulary. I can only describe the problem with common english words. Maybe someone can point me into the right direction and/or terms to look up. ...
0
votes
1answer
25 views

A class of “internal” endofunctors in cartesian closed categories

I'm interested in a class of endofunctors on cartesian closed categories with a quite natural definition, and am wondering whether/where this class has been studied so far (and how it is called). Fix ...
0
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0answers
41 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 ...
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0answers
29 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
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1answer
66 views

Is there a notion of property of a mathematical object?

I have an alphabet or set of symbols $\Sigma$ from which I can build sequences of symbols in $\Sigma^+$ (think of sentences of characters). Now I have a function ...
0
votes
2answers
51 views

Construction Types or Type Constructions?

In any (simple) type theory there are base types (i.e. the type of individuals and the type of propositions) and type builders (i.e. $\rightarrow$, which takes two types $t,t'$ and yields the type of ...
0
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0answers
20 views

Reexpressing the higher inductive-inductive surreals with only two mutual subdefinitions

In $\S 11.6$ of the HoTT book, the authors describe Conway's surreal numbers in terms of a higher inductive-inductive definition with three mutual subdefinitions. However, it seems to me that only ...
9
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0answers
162 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
2answers
137 views

Two questions on homotopy type theory

In reading the HoTT book, I have found that it is easy to become bogged down in detail and hard to tell the general 'big picture' of what is going on. I hope to get some general answer to the ...
6
votes
1answer
569 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 ...
1
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0answers
33 views

Proof that the empty type exists

Is there a proof that the empty type exists? If there is, it's not in my textbook and I haven't been able to find one online. I feel like if it's impossible to prove the existence of the empty type, ...
4
votes
3answers
95 views

What's a good resource to learn about the simply typed lambda calculus?

I've read An Introduction to Functional Programming Through Lambda Calculus by Greg Michaelson, and found it to be a very good resource to learn about the untyped lambda calculus. However, I want to ...
3
votes
1answer
170 views

Logic within Type theory. Is there a rough academic consensus on how this should be done?

I guess because of it's presence in several blogs on math and even physics, I've recently started to learn some type theory, often from people who know about it and from books which use it. I want to ...
5
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3answers
245 views

Introductory books as preparation to read Voevodsky homotopy-theory (HoTT) book

I would like to read Voevodsky HoTT book. However, I lack a lot of the basics. I would need a few introductory books first that cover topics like groupoids, fibrations, W -types, Homotopy theory. ...
2
votes
1answer
192 views

How to represent Smullyan's “Mockingbird” puzzles in (Homotopy) Type Theory?

(If you're unfamiliar with the puzzles from To Mock a Mockingbird, three pages tell you everything you should need.) Is it possible to solve the riddles in To Mock a Mockingbird in a "propositions as ...
4
votes
2answers
181 views

How to understand the definition of sets in homotopy type theory and the role of univalence?

Bear with me, I'm a physicist. In homotopy type theory, as I understand it, a type $X$ is a set if all the morphisms over its terms $x:X$ are identies. When I say "morphisms", then I view the term as ...
2
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0answers
46 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 ...
4
votes
1answer
85 views

How to describe free magmas in more structuralist terms?

Given a generating set $G$ (assume for simplicitly it consists entirely of urelements), the free magma on $G$ can be described concretely as follows. Its underlying set is the least $U \supseteq G$ ...
1
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1answer
42 views

Basic concerns about dependent function types

I am working my way through the introductory material of Homotopy Type Theory and by the end of section $1.5$ it is clear that I did not have the earlier material down as clearly as I had thought. I ...
5
votes
1answer
118 views

How does type theory handle division by zero and such?

Say I have a program that needs to not divide by zero: f(x): if nonzero(x): return sin(x)/x else: return 1 If we divide by zero, we get ...
3
votes
1answer
108 views

Give an example of a program in Simply Typed Lambda that produces Bottom.

I'm not sure how bottom applies to simply typed lambda calculus. not A is a common abbreviation for A -> ⊥ But I see no way to construct a function of that signature within the theory. Edit: A more ...
15
votes
3answers
592 views

What good is infinity?

I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent. When I was reading the comments in the video following (MF17), somebody ...
2
votes
1answer
109 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 ...
1
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2answers
88 views

Explanation of math statement

This symbols are used to describe left recursion : $A\to B\,\alpha\,|\,C$$B\to A\,\beta\,|\,D,$ It is taken from : http://en.wikipedia.org/wiki/Left_recursion How can these symbols be ...
1
vote
1answer
67 views

In type theory, why isn't $x = x' : X$ simply wrong?

If $X$ is a set, then personally, I tend to think of the equality relation on $X$ as a function $X^2 \rightarrow \mathrm{Bool}.$ Following this intuition, think that if $x$ and $y$ are variables of ...
4
votes
1answer
109 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
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1answer
87 views

Struggling with classes

I am still struggling with the concept of classes in ZF set theory. From Jech, Set Theory, p.5: That means: For every formula $\varphi(x)$ there is a class (which is definable). There are ...