# Questions tagged [realizability]

Realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them.

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### Which arithmetical sentences have no counterexamples in the sense of Kreisel?

It is a well-known fact that given a first-order sentence $\psi$ in prenex normal form $\forall x_1 \exists y_1 \forall x_2 \exists y_2 \dots \forall x_n \exists y_n \theta(x_1,\dots,x_n,y_1,\dots,y_n)...

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### Homotopical realizability

After a long story of dancing around the effective topos $ \mathcal{Eff}$, I finally resolved to get to the bottom of it. To this effect, working as it were backward, I ended up revisiting Kleene's ...

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### Realizability for constructive Zermelo-Fraenkel set theory

$ \def \CZF {\mathbf {CZF}}
\def \IZF {\mathbf {IZF}}
\def \A {\mathcal A}
\def \then {\mathrel \rightarrow}
\def \r {\mathrel \Vdash}
\DeclareMathOperator \V V $
In "Realizability for ...

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### Stack completions in realizability toposes

An internal category $\mathbb A$ in an elementary topos $\mathcal E$ is called an (intrinsic) stack if the indexed category that it represents, $(X\in \mathcal E) \mapsto \mathcal{E}(X,\mathbb A) \in \...

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### Are there simplicial spheres with "non-geometric symmetries"?

Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$.
...

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### Is there a polytope with an essentially unique shape?

More percisely:
Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations?
I suppose I have to assume that it has more than $d+2$ vertices/facets if ...

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### Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that
all vertices end up on a common sphere, and
the ...

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### Computability-theoretic results relevant to realizability

This may be a very naive question which only reflects my failure at literature search, but:
Although realizability (in its original form at least) is grounded in computability, the details of ...

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### Kleene realizability in Peano arithmetic

For completeness of MathOverflow and for clarity of the question, I will first recall a few things, including the the definition of Kleene realizability: experts can jump directly to the question ...

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### How is this HA unprovable formula recursive realizable?

In Realizability: A Historical Essay [Jaap van Oosten, 2002], it is said that recursive realizability and HA provability do not concur, because although every HA provable closed formula is realizable, ...

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### Did Kleene constructively prove Brouwer's axioms?

Harvey Friedman's request on the FoM-forum for an overview of current intuitionistic foundations revived the following question, which I have been meaning to ask for five years. (I'm no expert on ...

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### A polytime feasible subuniverse of the Effective Topos

The effective topos is a well known universe of sets suitable for abstract computability, as it is build "from the ground up" via the classical notion of realisability by Kleene.
I have found a few ...

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### Why is Kleene's notion of computability better than Banach-Mazur's?

In this post about the difference between the recursive and effective topos, Andrej Bauer said:
If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos ...