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Results tagged with lo.logic
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user 126667
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.
24
votes
Accepted
Can we disallow finite choice?
You might want topos theory. A topos is something like the category of sets, but the internal logic of a general topos is much weaker than ZF; it need not even be Boolean. An example of a topos is t …
- 25.2k
21
votes
Why do I find Category Theory mostly just a way to make simple things difficult?
As a topologist/category theorist with an interest in type systems I can assure you that I find pages full of sequents hard to understand :)
Actually I think the approaches are complementary. Suppos …
- 25.2k
12
votes
When are two proofs of the same theorem really different proofs
Some other answers have alluded to this, but just to spell it out explicitly: The Curry-Howard isomorphism, in one of its simpler forms, says that objects of the free cartesian closed category CCC[S] …
- 25.2k
10
votes
What is a logic?
I think there is supposed to be a correspondence between logics and kinds of category, e.g.,
(higher order?) classical logic
elementary topos with some extra properties?
(higher order?) in …
9
votes
What is a reference for an explicit, logic-based, statement of duality in category theory (i...
I would go even farther than the comments above, at least in the specific case you mention about computing (co)limits objectwise in a functor category. Once you know the statement for limits, deducin …
- 25.2k
6
votes
Accepted
What's the terminology for a sequent-like variant of category?
It sounds like you want the notion of a polycategory.
- 25.2k
5
votes
What is neutral constructive mathematics
I don't know the history of the phrase "neutral constructive mathematics", but I would guess that it is meant to be analogous to "neutral geometry". Neutral geometry (or absolute geometry) is the part …
- 25.2k
4
votes
Logical problems in category theory
I think you are just reading the wrong books. The most common solution to these problems is indeed Grothendieck universes. Really, these issues are not that big a deal, not because they are logicall …
- 25.2k
4
votes
Accepted
Naturally definable sets of natural numbers
Question 3: Can every finite set of natural numbers be defined by a natural formula?
If I understand correctly, the answer is yes: We can always rewrite $x = n_0 \vee x = n_1 \vee ... \vee x = n_ …
- 25.2k
0
votes
L^α_{β,γ}: do we need both α and β for model theory?
They are not the same when $\beta$ exceeds the cofinality of $\gamma$. In that case, we can form in $\mathcal{L}^{\infty}_{\beta,\gamma}$ a disjunction of existentials each of which binds fewer than $ …
- 25.2k