Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.

Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

We analyze three applications of Ramsey’s Theorem for 4-tuples to infinite traceable graphs and finitely generated infinite lattices using the tools of reverse mathematics. The applications in graph theory are shown to be equivalent to Ramsey’s Theorem while the application in lattice theory is shown to be provable in the weaker system RCA₀.

We show that the variety of equivalential algebras with regularization gives the algebraic semantics for the (↔, ¬¬)-fragment of intuitionistic propositional logic. We also prove that this fragment is hereditarily structurally complete.

Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on P_κ(λ) to the set of all a with is not λ⁺⁺-saturated (and even not -saturated in case 2^λ = λ⁺). We actually prove the stronger result that there is with |Q| = λ⁺⁺ such that A∩B is a non-cofinal subset of P_κ(λ) for any two distinct members A, B of Q, where NG_{κ, λ} denotes the game ideal on P_κ(λ). We also remark that for κ > ω₁, adding λ⁺³ Cohen subsets of ω₁ to makes NG_{κ, λ} λ⁺³-saturated.

Assuming certain conditions on a class of finitely generated first-order structures admitting the model-theoretical construction of a Fraïssé limit, we characterize retracts of such limits as algebraically closed structures in a class naturally related to . In this way we generalize an earlier description of retracts of the countably infinite random graph.

In this paper we define the forcing relation and prove its basic properties in the context of the theory ZFCA, i.e., ZFC minus the Foundation axiom and plus the Anti-Foundation axiom (AFA).

A quantified universe is a set M equipped with a Riesz space of real functions on Mⁿ, for each n, and a second order operation . Metric structures 4, graded probability structures 9 and many other structures in analysis are examples of such universes. We define ultraproduct of quantified universes and study properties preserved by this construction. We then discuss logics defined on the basis of classes of quantified universes which are closed under this construction.

Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A₁ = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ⁺ supercompact} is unbounded in κ. If in addition λ is 2^λ supercompact, then A₂ = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ⁺ supercompact} is unbounded in κ as well. The large cardinal hypotheses on λ are necessary, as we further demonstrate by constructing via forcing two distinct models in which either or . In each of these models, there is an indestructibly supercompact cardinal κ, and a restricted large cardinal structure above κ. If we weaken the indestructibility requirement on κ to indestructibility under partial orderings which are both κ-directed closed and (κ⁺, ∞)-distributive, then it is possible to construct a model containing a supercompact cardinal κ witnessing this degree of indestructibility in which every measurable cardinal δ < κ is (at least) δ⁺ supercompact.

We determine the large cardinal consistency strength of the existence of a λ-supercompact cardinal κ such that fails at λ. Indeed, we show that the existence of a λ-supercompact cardinal κ such that 2^λ ≥ θ is equiconsistent with the existence of a λ-supercompact cardinal that is also θ-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.

We study a natural subclass of continuous actions of Polish groups on Polish spaces which we call eventually open actions. We prove that this property characterizes the actions endowed with a complete system of hereditarily countable invariant structures introduced by Hjorth as a generalization of Scott sentences.

We study complexity of the index set of countably categorical theories and Ehrenfeucht theories in finite languages.

We study the structure of Σ¹₁ equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly i.e., Σ¹₁ but not equivalence classes. We also show the existence of incomparable Σ¹₁ equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ¹₁ equivalence relations (under both reducibilities) and show that existence of infinitely many properly Σ¹₁ equivalence classes that are complete as Σ¹₁ sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ¹₁ equivalence relations.

We give an example of a structure on the real line, and a manifold M definable in , such that M is definably connected but is not connected.