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Results tagged with lo.logic
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user 2811
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.
22
votes
Use of Conjectures to Prove a Theorem
How about this basic one:
Theorem: There exists an irrational number $p$ and an irrational number $q$ such that $p^q$ is rational.
Proof: Let $q=\sqrt{2}$ and note that it is irrational. Conjecture …
- 8,659
8
votes
1
answer
280
views
Ideals of statements?
The following is a somewhat vague question concerning logic, but with ideas from algebraic geometry (see in particular the example at the end). The vagueness is in the notion of "language".
Let $A$ …
- 8,659
11
votes
0
answers
305
views
What are the logical morphisms from a topos E to Set?
If $E$ is a topos, is there a nice way to characterize the category of logical morphisms $E\to Set$? Is it complete and/or cocomplete?
The topos $Set$ geometrically represents a point; what does it …
- 8,659
15
votes
3
answers
506
views
Conditions for a functor to induce a logical functor between presheaf toposes?
Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor between small categories.
Question: Under what conditions is the induced functor
$$F^*\colon\mathsf{Set}^\mathcal{D}\to\mathsf{Set}^\mathcal{C}$$ …
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11
votes
3
answers
479
views
Example of non-"propositional" local operators on a topos?
Let $\mathcal{E}$ be a topos, and let $\top\colon1\to\Omega$ be its subobject classifier. We refer to global elements $P\colon 1\to\Omega$ as propositions; they form a poset, denoted $(|\Omega|,\leq)$ …
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9
votes
1
answer
237
views
Topos properties from coverage conditions
For any category $C$ and coverage $J$ on it, let $\mathcal{E}:=\mathsf{Shv}(C,J)$ denote topos of sheaves on the site $(C,J)$. What sorts of results are known about the relationship between properties …
- 8,659
36
votes
2
answers
3k
views
What can be expressed in and proved with the internal logic of a topos?
The title of this post expresses what I really want, which is to learn how to wield the internal logic of a topos more effectively. However, to bring it down to earth, I'll ask a few basic questions a …
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8
votes
Reference request: Heyting algebra structure on Catalan numbers
I found a reference: "Dyck algebras, interval temporal logic and posets of intervals", which discusses these Heyting algebras (though not from a topos-theoretic perspective).
- 8,659
14
votes
1
answer
389
views
Reference request: Heyting algebra structure on Catalan numbers
I've noticed that for every natural number $n\in\mathbb{N}$, there is a finite Heyting algebra with cardinality $C(n)$, where $C(n)$ is the $n$th Catalan number,
$$1,1,2,5,14,42,132,\ldots$$
I'm looki …
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