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Results tagged with lo.logic
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user 49
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
6
votes
Variation on definition of logical functors avoiding power objects
I don't know of a notion of "logical functor" between predicative topoi that specializes automatically to the standard notion if they happen to be elementary topoi. But I do know a definition of "log …
- 66.8k
6
votes
Are there lightweight foundations for arbitrarily extendable objects?
I don't know enough about (1) or (3) to address them confidently, although I suspect that, as you suggested, some kind of internal type theory will do the trick. (Maybe ask a separate question about …
- 66.8k
10
votes
Accepted
Homotopy type theory: why are $0:\mathbb N$ and $\mathrm{succ}(0):\mathbb N$ not judgemental...
Daniel's answer is correct that the judgmental distinctness of $0$ and $\mathsf{succ}(m)$ is not what justifies a definition by pattern-matching. However, it is still a meaningful question of how to …
- 66.8k
7
votes
Accepted
Path types and identity types in dependent type theory
Prior to about a decade ago, no one used "path" terminology for identity types. The identification of the semantics of identity types with path objects dates to Awodey and Warren's Homotopy theoretic …
- 66.8k
14
votes
Is there a proof of strong normalisation that uses ordinal numbers?
I believe that Chapter 4 of Girard's Proofs and Types proves weak normalization for typed $\lambda$-calculus in this way, using ordinals up to $\omega^2$.
He first assigns a natural number "degree" to …
- 66.8k
6
votes
Accepted
Linear logic and linearly distributive categories
Yes, Cockett and Seely's comment about proof theory is a reference to the theory/category adjunction. Each kind of theory corresponds to a kind of category, for instance:
Theory
Category
simp …
- 66.8k
9
votes
Accepted
How to formulate the univalence axiom without universes?
One possibility along these lines is large eliminations for higher inductive types. For instance, here is a large elimination rule for the higher inductive interval type $\mathsf{I}$ with $0,1:\maths …
- 66.8k
5
votes
Accepted
Group objects via diagrams or generalized elements — Kripke–Joyal?
Sort of. Traditional Kripke–Joyal semantics can only force the truth of predicates, not the existence of structure. So once you have multiplication, unit, and inversion maps, you could use KJ semant …
- 12.9k
6
votes
Accepted
Internal language proof of Lawvere's fixed point theorem for cartesian closed categories
You're right that the statement of the theorem, and the entirety of the proof, don't fit inside the internal logic of a CCC. However, once given $f:B\to B$, the definition of $q$ and the proof that i …
- 66.8k
20
votes
Accepted
Coinduction for all?
This is a question that I've puzzled about myself, and I don't pretend to have The Answer. But here's one thought that I've found illuminating. Let's start by comparing the behavior of induction and …
CommunityBot
- 1
5
votes
Accepted
What are some interesting hyperdoctrines that are not classical models?
Maybe I am wrong, but it seems to me that the other answers are misunderstanding the question. The emphasis on syntactic hyperdoctrines seems to me beside the point.
A (classical, first-order) hyperd …
- 66.8k
16
votes
Taking a theorem as a definition and proving the original definition as a theorem
A Grothendieck topos can be defined either as the category of sheaves on a site, or as a category satisfying Giraud's axioms, or as an elementary topos that is bounded over $\rm Set$. I believe any o …
- 66.8k
14
votes
What is neutral constructive mathematics
I suppose I ought to contribute an answer to this. Maybe I should start by saying that I did not originate the term "neutral constructive mathematics". I believe I picked it up from Martin Escardo; …
- 66.8k
14
votes
Practical Benefits of HTT/univalent foundations for assisted proofs
You didn't specify exactly what "claimed benefits" for non-univalent type theory in general you're referring to, and I happen to believe that even non-univalent type theory does have substantial benef …
- 66.8k
10
votes
What is meant by a computational interpretation of univalence?
Roughly speaking, a type theory is computationally adequate if there is an algorithm that evaluates a term belonging to any type into a "normal form" of that type. The simplest form of this is when d …
- 66.8k