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Sets and Functions in Theoretical Physics

Abstract

It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue.

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Acknowledgments

We wish to express our deep gratitude to Alberto Levi, who raised relevant questions concerning a previous version of our separation schema in N theory. We have also benefitted enormously from insightful comments made by Newton da Costa and two anonymous referees for this journal. To all of them, our sincere thanks.

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Correspondence to Otávio Bueno.

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Sant’Anna, A.S., Bueno, O. Sets and Functions in Theoretical Physics. Erkenn 79, 257–281 (2014). https://doi.org/10.1007/s10670-013-9491-y

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  • DOI: https://doi.org/10.1007/s10670-013-9491-y

Keywords

  • Physical Theory
  • Axiomatic System
  • Virtual Particle
  • Explicit Mention
  • Primitive Concept