Document Type : Research
Abstract
Interactions between logic, measure and probability theories have always possessed significant importance in logic and model-theory. In this regard, numerous logical frameworks were introduced to connect these subjects. Integration-logic is amongst important ones of them that was first introduced by Keisler and Hoover and then developed in various works such as Bagheri-Pourmahdian paper and turned into a suitable logical framework for working with structures equipped with measures and integration operator. Also in a paper by Mofidi-Bagheri, a more abstract framework for working with operators more general than integration was introduced. Moreover, in a more recent work on connections of logic and measures, different aspects of dynamical-systems and measures in model-theory was published by Mofidi in 2018. One of the characteristics of Bagher-Pourmahdian framework is its boundedness, i.e. it is assumed that interpretation of every relation is a bounded function. Despite some advantages of this assumption (such as simplifying working with relations and proving ultraproduct and compactness theorems), it causes substantial limitations in the expressive-power of logic and its ability to interact with various mathematical structures. In this paper, we aim to resolve this limitations by strengthening and generalizing the framework of integration-logic in a way that relations be interpreted with (not-necessarily bounded) functions in L^p-spaces and furthermore, showing that fundamental results of ultraproduct and compactness theorems still hold (of course with new proofs and more subtle techniques). This generalization can provide more interactions with structures such as L^p-spaces and (not-necessarily-bounded) random-variables which are central notions in analysis and statistics.
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