arXiv subjects
Publications and source records attributed to Albert Visser.
In this paper, we show how to construct for a given consistent theory $U$ a $Σ^0_1$-predicate that both satisfies the Löb Conditions and the Kreisel Condition ---even if $U$ is unsound. We do this in such a way that $U$ itself can verify satisfaction of an internal version of the Kreisel Condition.
This paper from 2000 is a presentation of a status quæstionis at that tiime, to wit of the problem of the interpretability logic of {\em all}\/ reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
In this paper we study solution attempts for a problem posed by Ali Enayat: can there be a finitely axiomatized consistent sequential theory that interprets itself plus the (sentential or non-uniform) Tarski biconditionals? We provide a basic framework for the study of this question and discuss some solution attempts. We connect the question with some interesting conjectures. We briefly touch upon what happens if we consider uniform biconditionals.
Our paper is the first study of what one might call "reverse mathematics of explicit fixpoints". We study two methods of constructing such fixpoints for formulas whose principal connective is the intuitionistic Lewis arrow. Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allows several natural semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a well-understood modal system. The second one by de Jongh and Sambin, seemingly simpler, leads to a modal theory that proves harder to axiomatize in an elegant way. Apart from showing that both theories are incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixpoints of simple formulas. We also show that they are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfer to its finite extensions.
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 $S\inΣ_1$. 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 $Σ_1$-provability logic of Heyting Arithmetic.
The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable cut, then it shows that the property has a very small witness: i.e. a witness below a given standard number. We draw various consequences from the central result. For example (in rough formulations): (i) Every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative w.r.t. formulas of complexity $\leq n$. (ii) Every sequential model has, for any $n$, an extension that is elementary for formulas of complexity $\leq n$, in which the intersection of all definable cuts is the natural numbers. (iii) We have reflection for $Σ^0_2$-sentences with sufficiently small witness in any consistent restricted theory $U$. (iv) Suppose $U$ is recursively enumerable and sequential. Suppose further that every recursively enumerable and sequential $V$ that locally inteprets $U$, globally interprets $U$. Then, $U$ is mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measure depth of quantifier alternations.
In this paper, we provide a fairly general self-reference-free proof of the Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of Truth.
In this paper we provide a (negative) solution to a problem posed by Stanisław Krajewski. Consider a recursively enumerable theory U and a finite expansion of the signature of U that contains at least one predicate symbol of arity $\ge$ 2. We show that, for any finite extension $α$ of U in the expanded language that is conservative over U, there is a conservative extension $β$ of U in the expanded language, such that $α\vdashβ$ and $β\nvdashα$. The result is preserved when we consider either extensions or model-conservative extensions of U in stead of conservative extensions. Moreover, the result is preserved when we replace $\vdash$ as ordering on the finitely axiomatized extensions in the expanded language by a special kind of interpretability, to wit interpretability that identically translates the symbols of the U-language. We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.
C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.