The algebraic category is the image of MV, the category whose objects are the MV-algebras, by the equivalence (cf. ). In this paper we define the logic Ł^{•} whose Lindenbaum algebra is an MV-algebra (object of ), and establish a link between Ł^{•} and the infinite valued Łukasiewicz logic Ł. We define U-operators, that have properties of universal quantifiers, and establish a bijection that maps an MV-algebra endowed with a U-operator (cf. ) into an MV^{•}-algebra endowed with a U-operator. This map extends to a functor that is a categorical equivalence.

This work addresses a basic question by Kunen: how many normal measures can there be on the least measurable cardinal? Starting with a measurable cardinal κ of Mitchell order less than two () we define a Prikry type forcing which turns the number of normal measures over κ to any while making κ the first measurable.

]]>We study a closed unbounded self-adjoint operator *Q* acting on a Hilbert space *H* in the framework of *Metric Abstract Elementary Classes* (MAECs). We build a suitable MAEC for such a structure, prove it is ℵ_{0}-categorical and ℵ_{0}-stable up to a system of perturbations. We give an explicit continuous axiomatization for the class. We also characterize non-splitting and show it has the same properties as non-forking in superstable first order theories. Finally, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.

In this paper we first describe the Priestley duality for pseudocomplemented De Morgan algebras by combining the known dualities of distributive p-algebras due to Priestley and for De Morgan algebras due to Cornish and Fowler. We then use it to characterize congruence-permutability, principal join property, and the property of having only principal congruences for pseudocomplemented De Morgan algebras. The congruence-uniform pseudocomplemented De Morgan algebras are also described.

]]>Suppose is a triple of two theories in vocabularies with cardinality λ, and a τ_{1}-type *p* over the empty set that is consistent with *T*_{1}. We consider the Hanf number for the property “there is a model *M*_{1} of *T*_{1} which omits *p*, but is saturated”. In [2], we showed that this Hanf number is essentially equal to the Löwenheim number of second order logic. In this paper, we show that if *T* is superstable, then the Hanf number is less than .

We consider the complexity of satisfiability in ε-logic, a probability logic. We show that for the relational fragment this problem is -complete for rational , answering a question by Terwijn. In contrast, we show that satisfiability in 0-logic is decidable. The methods we employ to prove this fact also allow us to show that 0-logic is compact, while it was previously shown that ε-logic is not compact for .

]]>We use techniques due to Moti Gitik to construct models in which for an arbitrary ordinal ϱ, is both the least measurable and least regular uncountable cardinal.

]]>We introduce and analyse a theory of finitely stratified general inductive definitions over the natural numbers, , and establish its proof theoretic ordinal, . The definition of bears some similarities with Leivant's ramified theories for finitary inductive definitions.

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