相关论文: On the Realizability of Prime Conjectures in Heyti…
This work introduces a novel framework of uniform realizability that unifies and generalizes various realizability interpretations of logic, particularly focussing on the treatment of atomic formulas and quantifiers. Traditional…
For the Heyting Arithmetic HA, HA* is defined as the theory $\{A\mid {\sf HA}\vdash A^{\Box}\}$, where $A^{\Box}$ is called the box translation of $A$. We characterize the $\Sigma_1$-provability logic of HA* as a modal theory ${\sf…
In this paper we provide a semantic and syntactic analysis of parametrised natural numbers object in coherent categories, or pr-coherent categories. Semantically, we show the definable functions in the initial pr-coherent category are…
It is well known that the graph of a total $\mathbf{\Sigma}^1_n$-function is $\mathbf{\Pi}^1_n$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total…
For any partial combinatory algebra (PCA for short) A, the class of A-representable partial functions from N to A quotiented by the filter of cofinite sets of N, is a PCA such that the representable partial functions are exactly the…
A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and…
The Sigma formulas of the language of arithmetic express semidecidable relations on the natural numbers. More generally, whenever a totality of objects is regarded as incomplete, the Sigma formulas express relations that are witnessed in a…
We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete $\Pi^1_1$ set…
It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<\sigma<1$; this is certainly not true for $\sigma>1$. Answering a question of Bombieri and Ghosh, we give a simple…
In this paper we introduce a modal theory $H_{\sigma}$, which is sound and complete for arithmetical $\Sigma$_1 substitutions in ${\bf HA}$, in other words, we will show that $H_{\sigma}$ is the $\Sigma$_1-provability logic of ${\bf HA}$.…
Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of…
Skolem functions play a central role in the study of first order logic, both from theoretical and practical perspectives. While every Skolemized formula in first-order logic makes use of Skolem constants and/or functions, not all such…
We present a new set of reductions for derivations in natural deduction that can extract witnesses from closed derivations of simply existential formulas in Heyting Arithmetic (HA) plus the Excluded Middle Law restricted to simply…
We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$…
Ordinal analysis induces a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the…
We apply to the semantics of Arithmetic the idea of ``finite approximation'' used to provide computational interpretations of Herbrand's Theorem, and we interpret classical proofs as constructive proofs (with constructive rules for $\vee,…
We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is…
It is well known that whenever a class of structures $\mathcal{K}_1$ is interpretable in a class of structures $\mathcal{K}_2$, then the hereditary undecidability of (a fragment of) the theory of $\mathcal{K}_1$ implies the hereditary…
Timothy Carlson's patterns of resemblance employ the notion of $\Sigma_1$-elementarity to describe large computable ordinals. It has been conjectured that a relativization of these patterns to dilators leads to an equivalence with…
We examine the interplay between projectivity (in the sense that was introduced by S.~Ghilardi) and uniform post-interpolant for the classical and intuitionistic propositional logic. More precisely, we explore whether a projective…