Mirna Dzamonja | Searcharxiv

arXiv2023

The Lindstrom's Characterizability of Abstract Logic Systems for Analytic Structures Based on Measures

In 1969, Per Lindstrom proved his celebrated theorem characterising the first-order logic and established criteria for the first-order definability of formal theories for discrete structures. K. J. Barwise, S. Shelah, J. Vaananen and others extended Lindstrom's characterizability program to classes of infinitary logic systems, including a recent paper by M. Dzamonja and J. Vaananen on Karp's chain logic, which satisfies interpolation, undefinability of well-order, and is maximal in the class of logic systems with these properties. The novelty of the chain logic is in its new definition of satisfability. In our paper, we give a framework for Lindstrom's type characterizability of predicate logic systems interpreted semantically in models with objects based on measures (analytic structures). In particular, Hajek's Logic of Integral is redefined as an abstract logic with a new type of Hajek's satisfiability and constitutes a maximal logic in the class of logic systems for describing analytic structures with Lebesgue integrals and satisfying compactness, elementary chain condition, and weak negation.

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arXiv2008

On $D$-spaces and Discrete Families of Sets

We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving discrete families and the well known question of the existence of families allowing partial transversals without having a transversal themselves, and use it to give non-trivial instances of the incompactness phenomenon in the context of discretisations.

math.LO↗