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Zenith Grant Awardee

Donald Spector

Hobart and William Smith Colleges

Project Title

Set Theoretic Forcing and Information Theory

Project Summary

Physics uses numbers that can in principle be specified to infinite precision. There is a well developed theory of information that incorporates key ideas from physics, and there are arguments suggesting that, at a fundamental level, the theories of physics should be theories of information, yet the existing mathematical correspondence of physics and information breaks down when we consider numbers specified to infinite precision. Mathematicians, in dealing with the surprising finding that there are different sizes of infinities, have developed a technique known as forcing. In my research, I will demonstrate that the mathematical technique of forcing provides the tools necessary to characterize how much information is encoded in arbitrarily well specified numbers. The techniques of forcing will provide a way to understand how experiments can squeeze out more information by going to every higher precision, how theoretical physics can track the information flow from input to output, and how a theory of information can be developed that works whether we have a finite or an infinite number of possible outcomes. The resulting, more powerful theory of the relationship between physics and information would also represent the first real-world application of the abstract mathematical technique of forcing.

Technical Abstract

Theories of physics, because they are defined over the real numbers, involve the specification of infinite – even uncountably infinite – amounts of information. Yet existing characterizations of information are ill suited to systems with an infinity of possible values. I propose to address the question of how to measure the information in cases where the number of possibilities is infinite by using the set theoretic technique of forcing. Note that a chaotic system, despite its sensitive dependence on initial conditions, behaves physically in some particular way, indicating its ability somehow to track its initial state to infinite precision. Similarly, scientific measurements can never obtain the precision of a real number, but can, in principle, be refined arbitrarily well. I will use forcing to mathematize the notion of information in such contexts, and in a way that offers a useful characterization of information when an infinite number of outcomes is possible. Different forcing methods will provide insights into different ways in which information is embedded in continuous systems, with the associated notions of generic real numbers that arise in forcing constructions extending notions inherent in maximum entropy methods and in the Asymptotic Equipartition Theorem of conventional information theory.

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