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A maximality theorem for continuous first order theories

Ackerman Nathanael, Karker Mary Leah

Reports on Mathematical Logic

2022

Vol. 57

61--93

In this paper we prove a Lindstr¨om like theorem for the logic consisting of arbitrary Boolean combinations of first order sentences. Specifically we show the logic obtained by taking arbitrary, possibly infinite, Boolean combinations of first order sentences in countable languages is the unique maximal abstract logic which is closed under finitary Boolean operations, has occurrence number ω1, has the downward L¨owenheim-Skolem property to ω and the upward L¨owenheim-Skolem property to uncountability, and contains all complete first order theories in countable languages as sentences of the abstract logic. We will also show a similar result holds in the continuous logic framework of [5], i.e. we prove a Lindstr¨om like theorem for the abstract continuous logic consisting of Boolean combinations of first order closed conditions. Specifically we show the abstract continuous logic consisting of arbitrary Boolean combinations of closed conditions is the unique maximal abstract continuous logic which is closed under approximate isomorphisms on countable structures, is closed under finitary Boolean operations, has occurrence number ω1, has the downward L¨owenheim-Skolem property to ω, the upward L¨owenheim-Skolem property to uncountability and contains all first order theories in countable languages as sentences of the abstract logic.