Chapter XIX: Between number theory and set theory

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This chapter discusses a number of interconnections between number theory, that is, the theory of natural numbers or nonnegative integers and certain systems of set theory. Set theory is equivalent to number theory. If infinite sets are allowed but not impredicative sets, then set theory still resembles number theory quite well. The big gap between number theory and set theory proper is the introduction of impredicative sets in set theory. The chapter formulates these connections between number theory and set theory. The chapter reviews the general set theory or Zermelo's set theory minus the axiom of infinity. The system G is the basic system of general set theory including the axiom of extensionality, the Aussonderungs axiom, and an axiom assuring the existence of finite sets. The system Z is the ordinary system of elementary number theory. © 1963, Science Press

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