The type-theory tag has no usage guidance.

**28**

votes

**6**answers

5k views

### How true are theorems proved by Coq?

Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...

**4**

votes

**1**answer

636 views

### What fails when using call/cc as realizer of the Peirce formula

Define the axiom constants $p_{A,B}^{((A\rightarrow B)\rightarrow A)\rightarrow A}$ as realizers of the Peirce formula, and $f_A^{\bot\rightarrow A}$ as realizers of the Ex Falso Quodlibet. Then $p_{A\...

**-1**

votes

**5**answers

814 views

### finding cutting edge papers and books

Hi all,
what are the best strategies to find cutting edge papers and books on a field of mathematics?
..
Example:
2-3 years ago I had to analyze a time series. I found a paper and showed that to ...

**5**

votes

**3**answers

611 views

### What do I call type theory without Curry-Howard?

Is there a word I can say which will convey to type theorists that I am not thinking about types as propositions?
Background: as a category theorist, I am mostly interested in type theories as a ...

**4**

votes

**3**answers

518 views

### Can a typing judgment admit essentially different derivations?

In any sort of type theory, there are a bunch of rules for constructing derivations of typing judgments such as $x:A,\; y:B(x) \;\vdash\; z:C(x,y)$. (I intend to include also judgments of the form $B:...

**3**

votes

**0**answers

195 views

### An elegant formulation for typed sets

Fix a poset $T$, which we'll think of as a set of "types," interpreting $a \leq b$ as "$a$ is more general than $b$." Construct a category of TSet as follows.
Objects: Pairs ($X$, $\tau : X \...

**2**

votes

**1**answer

252 views

### Relation between different definitions of types [closed]

Is there any connection between the definition of type in model theory and the definitions from type theory? Is there any explanation why the same term is used for these notions, maybe in the ...

**6**

votes

**4**answers

715 views

### What is the intuitive meaning of star and box in a pure type system?

The systems of the λ-cube have the axiom $\star:\square$.
I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and $\...

**1**

vote

**2**answers

992 views

### What is a semigroup or, what do I do with that associativity proof?

Mathematically, I know what a semigroup is: It is a set S along with an associative binary operation $* : S \times S \rightarrow S$. So far, so good.
From a computational perspective, one can ...

**11**

votes

**4**answers

901 views

### Reference request for type theory

I am interested in learning the theory of types, especially in how they can provide a foundation to mathematics different to sets and how they can avoid self-referential paradoxes by stipulating that ...

**12**

votes

**2**answers

2k views

### What is the manner of inconsistency of Girard's paradox in Martin Lof type theory

I am aware that assigning the type of Type to be Type (rather than stratifying to a hierarchy of types) leads to an inconsistency. But does this inconsistency allow the construction of a well-typed ...

**7**

votes

**4**answers

1k views

### Can dependent sums be encoded as dependent products?

Please forgive any unorthodox notation or obvious errors here... I'm trying to get an intuition for dependently typed languages, so I'm starting out by seeing which analogies I can take from the ...

**4**

votes

**1**answer

506 views

### What notions of universe does predicative type theory admit?

Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in ...

**6**

votes

**2**answers

682 views

### Recursively dependent types?

Is there such a thing as "recursively dependent types"? Specifically, I would like a dependent type theory containing a type $A(x)$ which depends on a variable $x: A(z)$, where $z$ is a particular ...

**13**

votes

**7**answers

2k views

### What is lambda calculus related to?

So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based.
I was wondering if anyone had a suggestion ...