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Area of Science:

  • Probability Theory
  • Mathematical Philosophy

Background:

  • Classical probability theory faces limitations in combining regularity with perfect additivity.
  • Infinitesimal probabilities have faced philosophical objections.

Purpose of the Study:

  • To introduce Non-Archimedean Probability (NAP) theory, offering a framework that combines regularity with perfect additivity.
  • To explore the philosophical motivations behind the chosen axioms for NAP.
  • To address and refute philosophical objections raised against infinitesimal probabilities.

Main Methods:

  • Axiomatic development of Non-Archimedean Probability (NAP) theory.
  • Analysis of continuity and conditional probability within the NAP framework.
  • Definition of NAP functions through infinite sums.
  • Comparison with numerosity theory.
  • Examination of objections using examples like infinite lotteries and coin tosses.

Main Results:

  • NAP theory provides a robust alternative to classical probability, allowing for perfect additivity and regularity.
  • Philosophical objections to infinitesimal probabilities are systematically addressed and countered.
  • The theory demonstrates connections to measure theory, utility, credence, and chance.

Conclusions:

  • Non-Archimedean probability functions offer a philosophically motivated and mathematically sound extension of probability theory.
  • The NAP framework successfully integrates infinitesimal concepts, overcoming previous objections.
  • The theory has implications for understanding regularity, uniformity, and conditional probabilities.