Abstract
The axioms of infinity are justified by two general principles: principle of transition from potential to actual infinity and principle of existence of singular sets. An early application of the first principle is Dedekind's argument the existence of infinite sets. Also, the first principle allows formulating the axiom of inaccessible numbers and Lēvy's reflection schema for Zermelo–Fraenkel set theory. The second principle allows formulating still stronger axioms and is applied in the formulation of the axiom stating the existence of measurable cardinals. The chapter presents some statements related to the principle of existence of singular sets, which are equivalent to reflection principles, and discusses Lēvy's axiom that postulates the existence of at least one inaccessible number in the range of every normal function defined for all ordinals. Some new axioms of strong affinity, various theorem, and lemmas are also discussed in the chapter.
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