This paper investigates a first-order extension of GL called \. We outline briefly the history that led to \, its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \ is arithmetically complete. As expanded below, \ is a first-order modal logic (...) that along with its built-in ability to simulate general classical first-order provability―"\" simulating the the informal classical "\"―is also arithmetically complete in the Solovay sense. (shrink)

We introduce a Gentzen-style modal predicate logic and prove the cut-elimination theorem for it. This sequent calculus of cut-free proofs is chosen as a proxy to develop the proof-theory of the logics introduced in [14, 15, 4]. We present syntactic proofs for all the metatheoretical results that were proved model-theoretically in loc. cit. and moreover prove that the form of weak reflection proved in these papers is as strong as possible.

A well established technique toward developing the proof theory of a Hilbert-style modal logic is to introduce a Gentzen-style equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1, 6, 8, 18, 10, 12]). In the first-order modal case, on one hand we know that the Gentzenisation of the straightforward first-order extension of GL, the logic QGL, admits no cut elimination (if the rule is included as primitive; (...) or, if not included, then the rule is not admissible [1]). On the other hand the (cut-free) Gentzenisations of the first-order modal logics M3 and ML3 of [10, 12] do have cut as an admissible rule. The syntactic cut admissibility proof given in [18] for the Gentzenisation of the propositional provability logic GL is extremely complex, and it was the basis of the proofs of cut admissibility of the Gentzenisations of M3 and ML3, where the presence of quantifiers and quantifier rules added to the complexity and length of the proof. A recent proof of cut admissibility in a cut-free Gentzenisation of GL is given in [5] and is quite short and easy to read. We adapt it here to revisit the proofs for the cases of M3 and ML3, resulting to similarly short and easy to read proofs, only slightly complicated by the presence of quantification and its relevant rules. (shrink)

We introduce a first order extension of GL, called ML 3 , and develop its proof theory via a proxy cut-free sequent calculus GLTS. We prove the highly nontrivial result that cut is a derived rule in GLTS, a result that is unavailable in other known first-order extensions of GL. This leads to proofs of weak reflection and the related conservation result for ML 3 , as well as proofs for Craig’s interpolation theorem for GLTS. Turning to semantics we prove (...) that ML 3 is sound with respect to arithmetical interpretations and that it is also sound and complete with respect to converse well-founded and transitive finite Kripke models. This leads us to expect that a Solovay-like proof of arithmetical completeness of ML 3 is possible. (shrink)

Reference [12] introduced a novel formula to formula translation tool that enables syntactic metatheoretical investigations of first-order modallogics, bypassing a need to convert them first into Gentzen style logics in order torely on cut elimination and the subformula property. In fact, the formulator tool,as was already demonstrated in loc. cit., is applicable even to the metatheoreticalstudy of logics such as QGL, where cut elimination is unavailable. This paper applies the formulator approach to show the independence of the axiom schema ☐A (...) → ☐∀ A of the logics M3and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics. (shrink)

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume II, on formal (ZFC) set theory, incorporates a self-contained 'chapter 0' on proof techniques so (...) that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques will provide the reader with a solid foundation in set theory and provides a context for the presentation of advanced topics such as absoluteness, relative consistency results, two expositions of Godel's constructible universe, numerous ways of viewing recursion, and a chapter on Cohen forcing. (shrink)

We refine the definition of II-computability of [12] so that oracles have a “consistent”, but natural, behaviour. We prove a Kleene Normal Form Theorem and closure of semi-recursive relations under ∃1. We also show that in this more inclusive computation theory Post's theorem in the arithmetical hierarchy still holds.

In addition, this book contains tools that, in principle, can search a set of algorithms to see whether a problem is solvable, or more specifically, if it can be solved by an algorithm whose computations are efficient.