Borel Structures for First-order and Extended Logics

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This chapter focuses on Borel structures for first-order and extended logics. The aspects of model theory to be discussed in this chapter blend together two general problems in model theory. The first, that of finding and examining “natural” logics that are more expressive than first-order logic, has been an active and stimulating area of research for at least the past twenty-five years. Work in this area has typically operated between two constraints: that the logic should express concepts beyond first-order logic that are mathematically significant and also that which is imposed by Lindstrom's theorem—i.e., some of the properties of first-order logic that render it so manageable must be sacrificed. The second broad problem concerns avoiding pathology that might creep in when a particular set of axioms is studied. All theories discussed in this chapter are constructed from a countable vocabulary τ. The chapter considers first-order logic and finitary extensions of it obtained by the addition in different combinations of the quantifiers Q, Q c , and Q m . Some first-order extensions of Borel model theory are also discussed. The most basic theorem in the theory is Borel completeness theorem. The progress that has been made in the study of the logics obtained by adjoining various combinations of Q, Q c , and Q m to first-order logic has been described.