Provability Logic and the Completeness Principle

In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that prove the schemes $A\to\triangle A$ and $\Box\triangle S\to\Box S$ for $S\in\Sigma_1$. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the $\Sigma_1$-provability logic of Heyting Arithmetic

On the Existence of Pushouts of Realizability Toposes

We consider two preorder-enriched categories of ordered PCAs: $\mathsf{OPCA}$, where the arrows are functional morphisms, and $\mathsf{PCA}$, where the arrows are applicative morphisms. We show that $\mathsf{OPCA}$ has small products and finite biproducts, and that $\mathsf{PCA}$ has finite coproducts, all in a suitable 2-categorical sense. On the other hand, $\mathsf{PCA}$ lacks all nontrivial binary products. We deduce from this that the pushout, over $\mathsf{Set}$, of two nontrivial realizability toposes is never a realizability topos.Comment: 19 pages; revised argument in Section 6, added remarks and reference

Internal Partial Combinatory Algebras and their Slices

A partial combinatory algebra (PCA) is a set equipped with a partial binary operation that models a notion of computability. This paper studies a generalization of PCAs, introduced by W. Stekelenburg, where a PCA is not a set but an object in a given regular category. The corresponding class of categories of assemblies is closed both under taking small products and under slicing, which is to be contrasted with the situation for ordinary PCAs. We describe these two constructions explicitly at the level of PCAs, allowing us to compute a number of examples of products and slices of PCAs. Moreover, we show how PCAs can be transported along regular functors, enabling us to compare PCAs constructed over different base categories. Via a Grothendieck construction, this leads to a (2-)category whose objects are PCAs and whose arrows are generalized applicative morphisms. This category has small products, which correspond to the small products of categories of assemblies, and it has finite coproducts in a weak sense. Finally, we give a criterion when a functor between categories of assemblies that is induced by an applicative morphism has a right adjoint, by generalizing the notion of computational density introduced by P. Hofstra and J. van Oosten

Computability Models and Realizability Toposes

A partial combinatory algebra (PCA) is a model of computation that embodies a certain notion of algorithm. The elements of a PCA act simultaneously as algorithms and as inputs that may be fed to these algorithms. For each PCA, we can construct a model of intuitionistic mathematics, called the realizability topos of the PCA. In this model, validity is governed by its underlying PCA: the truth of a statement must be corroborated by an algorithm of the PCA. In this thesis, we investigate some general properties of the construction that assigns, to a PCA, its corresponding realizability topos. We show that realizability toposes are not closed under certain elementary operations on categories, such as products and slices. On the basis of these negative results, we introduce natural classes of ‘realizability-like’ toposes that are closed under these operations. In addition, we investigate the general theory of computing with functions on a PCA. This includes computation relative to an oracle, which is a form of computation that may consult an external resource (the oracle) a finite number of times before coming up with a final output. Moreover, we study computation with higher-order functionals, which are functions that take functions, rather than elements, as their input

On (co)products of partial combinatory algebras, with an application to pushouts of realizability toposes

We consider two preorder-enriched categories of ordered partial combinatory algebras: OPCA, where the arrows are functional (i.e., projective) morphisms, and OPCA†, where the arrows are applicative morphisms. We show that OPCA has small products and finite biproducts, and that OPCA† has finite coproducts, all in a suitable 2-categorical sense. On the other hand, OPCA† lacks all nontrivial binary products. We deduce from this that the pushout, over Set, of two nontrivial realizability toposes is never a realizability topos. In contrast, we show that nontrivial subtoposes of realizability toposes are closed under pushouts over Set

Third-order functionals on partial combinatory algebras

Computability relative to a partial function f on the natural numbers can be formalized using the notion of an oracle for this function f. This can be generalized to arbitrary partial combinatory algebras, yielding a notion of `adjoining a partial function to a partial combinatory algebra A'. A similar construction is known for second-order functionals, but the third-order case is more difficult. In this paper, we prove several results for this third-order case. Given a third-order functional Φ on a partial combinatory algebra A, we show how to construct a partial combinatory algebra A[Φ] where Φ is `computable', and which has a `lax' factorization property. Moreover, we show that, on the level of first-order functions, the effect of making a third-order functional computable can be described as adding an oracle for a first-order function

Provability Logic and the Completeness Principle

In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that prove the schemes $A\to\triangle A$ and $\Box\triangle S\to\Box S$ for $S\in\Sigma_1$. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the $\Sigma_1$-provability logic of Heyting Arithmetic

Provability logic and the completeness principle

The logic iGLC is the intuitionistic version of Löb's Logic plus the completeness principle A→□A. In this paper, we prove an arithmetical completeness theorems for iGLC for theories equipped with two provability predicates □ and △ that prove the schemes A→△A and □△S→□S for S∈Σ 1 . We provide two salient instances of the theorem. In the first, □ is fast provability and △ is ordinary provability and, in the second, □ is ordinary provability and △ is slow provability. Using the second instance, we reprove a theorem previously obtained by Mohammad Ardeshir and Mojtaba Mojtahedi [1] determining the Σ 1 -provability logic of Heyting Arithmetic

Provability logic and the completeness principle

The logic iGLC is the intuitionistic version of Löb's Logic plus the completeness principle A→□A. In this paper, we prove an arithmetical completeness theorems for iGLC for theories equipped with two provability predicates □ and △ that prove the schemes A→△A and □△S→□S for S∈Σ 1 . We provide two salient instances of the theorem. In the first, □ is fast provability and △ is ordinary provability and, in the second, □ is ordinary provability and △ is slow provability. Using the second instance, we reprove a theorem previously obtained by Mohammad Ardeshir and Mojtaba Mojtahedi [1] determining the Σ 1 -provability logic of Heyting Arithmetic

Provability Logic and the Completeness Principle

In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that prove the schemes $A\to\triangle A$ and $\Box\triangle S\to\Box S$ for $S\in\Sigma_1$. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the $\Sigma_1$-provability logic of Heyting Arithmetic