A Liouville function is an analytic function $H: \C \rightarrow \C$ with a Taylor series $\sum_n=1^\infty xⁿ/a_n$ such the a_n’s form a “very fast growing” sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H.

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality \aleph₀. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ⁺⁺ graphs on μ⁺ that taken jointly are universal for the graphs on μ⁺, while $2^μ+ \gg μ⁺⁺$. The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic $\simL$ in the diamond language with each logic $\La$ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

Craig’s interpolation lemma (if $φ\rightarrowψ$ is valid, then $φ\rightarrowθ$ and $θ\rightarrowψ$ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.

We reprove Gitik’s theorem that if the GCH holds and $o(\gk)=\gk+1$ then there is a generic extension in which $\gk$ is still measurable and there is a closed unbounded subset C of $\gk$ such that every $ν\in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik, the iterated forcing $\radin_\gl+1$ used in this paper has the property that if $\gl$ is a cardinal less then $\gk$ then $\radin_\gl+1$ can be factored in V as $\radin_\gk+1=\radin_\gl+1\times\radin_\gl+1,\gk$ where $\card{\radin_\gl+1}\le\gl⁺$ and $\radin_\gl+1,\gk$ does not add any new subsets of $\gl$.

We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension $V^\P$ and all subsequent extensions V^\P*\Qdot holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\possible\necessaryφ)\implies\necessaryφ$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with \ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $V_δ\elesub V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that $\ORD$ is Mahlo. The strongest principle along these lines is $\necessary\MPtilde$, which asserts that $\MPtilde$ holds in V and all forcing extensions. From this, it follows that 0^# exists, that x^# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c =(a \cup c) \cap (b\cupc), a \cup c | b \cup c, and c a \cup b.

We characterize the classifiability of a countable first-order theory T in terms of the solvability (in the sense of [Friedman00]) of the potential-isomorphism problem for models of T.

We show that for any infinite cardinal κ, every strongly (κ+1)-strategically closed poset is strongly κ⁺-strategically closed if and only if AP_κ (the approachability property) holds, answering the question asked in [k5]. We also give a complete classification of strengths of strategic closure properties and that of strong strategic closure properties respectively.

We define a nontrivial variety of boolean algebras with operators such that every member of the variety is atomless. This shows that not every variety of boolean algebras with operators is generated by its atomic members, and thus establishes a strong incompleteness result in (multi-)modal logic.

We show that Blackwell determinacy in $\Ll(\R)$ implies determinacy in $\Ll(\R)$.

We study automorphism groups of trivial strongly minimal structures. First we give a characterization of structures of bounded valency through their groups of automorphisms. Then we characterize the triplets of groups which can be realized as the automorphism group of a non algebraic component, the subgroup stabilizer of a point and the subgroup of strong automorphisms in a trivial strongly minimal structure, and also we give a reconstruction result. Finally, using HNN extensions we show that any profinite group can be realized as the stabilizer of a point in a strongly minimal structure of bounded valency.

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [A97], due to the first author.

We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta’s theory of truth over arithmetic using \emph{fully varied revision} sequences yields a complete \Pi¹₃ set of integers; (iii) the set of \emph{stably categorical} sentences using their revision operator ψ is similarly \Pi¹₃ and which is complete in Gödel’s universe of constructible sets L; (iv) give an alternative account of a theory of truth—realistic variance that simplifies full variance, whilst at the same time arriving at Kripkean fixed points.