Related papers: An Analysis of Tennenbaum's Theorem in Constructiv…
We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…
We develop synthetic notions of oracle computability and Turing reducibility in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. As usual in synthetic approaches, we employ a…
It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model.…
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega$ and that non-standard models of true arithmetic must have Scott rank greater than…
We construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolutions, would contradict the Church-Turing thesis which lies at the foundation of computer…
General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might…
The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two…
When can a model of a physical system be regarded as computable? We provide the definition of a computable physical model to answer this question. The connection between our definition and Kreisel's notion of a mechanistic theory is…
Constructive arithmetic, or the Markov arithmetic MA, is obtained from intuitionistic arithmetic HA by adding the following two principles: the Markov principle M which distinguishes constructivism from intuitionism, and the so-called…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become…
Orbit-finite models of computation generalise the standard models of computation, to allow computation over infinite objects that are finite up to symmetries on atoms, denoted by $\mathbb{A}$. Set theory with atoms is used to reason about…
In the early 1970's S.Tennenbaum proved that all countable models of PA^- + forall_1-Th(N) are embeddable into the reduced product N^omega/F, where F is the cofinite filter. In this paper we show that if M is a model of PA^- +…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on G\"{o}del's reformulation, and including additional details…
Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the…
We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…
We show that Church's thesis, the axiom stating that all functions on the naturals are computable, does not hold in the cubical assemblies model of cubical type theory. We show that nevertheless Church's thesis is consistent with univalent…
Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally…
This article presents a computational semantics for classical logic using constructive type theory. Such semantics seems impossible because classical logic allows the Law of Excluded Middle (LEM), not accepted in constructive logic since it…