Logical Studies

Logic, measures and unbounded integration logic

Document Type : Research

https://doi.org/10.30465/lsj.2022.38229.1375
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.

Keywords

Unbounded integration logic
ultraproduct theorem
compactness theorem
L^p-spaces
Bagheri S.M, Pourmahdian M. (2009), The logic of integration, Arch. Math. Logic 48:465-492.
 
Fajardo S., Keisler H.J. (2002), Model theory of stochastic processes, Lecture Notes in Logic 14 ASL.
 
Folland G.B (1999), Real analysis, Modern techniques and their applications, second edition.
 
Hoover D.N (1978), Probability logic, Annals of Mathematical Logic 14, 287-313.
 
Keisler H.J. (1985), Probability quantifiers, in Model Theoretic Logic, edited by J. Barwise and S. Feferman, Springer-Verlag.
 
Mofidi, A. (2018), On some dynamical aspects of NIP theories, Arch. Math. Logic 57 (1-2) 37–71.
 
Mofidi, A. (2020), On partial cubes, well-graded families and their duals with some applications in graphs, Discrete Appl. Math, 283, 207–230.
 
Mofidi A, Bagheri S.M (2011), Quantified universes and ultraproduct, math. logic quarterly.
 
 
Logical Studies
Volume 12, Issue 2
September 2021
Pages 235-250

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