The Lindstrom's Characterizability of Abstract Logic Systems for Analytic Structures Based on Measures
This work addresses foundational issues in mathematical logic for researchers, but it is incremental as it builds on existing extensions of Lindstrom's theorem to new domains.
The paper tackles the problem of extending Lindstrom's characterizability program to predicate logic systems for analytic structures based on measures, resulting in a framework that redefines Hajek's Logic of Integral as a maximal logic in this class with properties like compactness and weak negation.
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.