The cardinality of the set of all ordinal numbers is not defined within Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) because the class of all ordinals constitutes a proper class rather than a set. This distinction arises from the Burali-Forti paradox, which illustrates that there are too many ordinals to encapsulate within a single set. Consequently, it is impossible to assign a cardinality to a proper class, making the question of the cardinality of all ordinals nonsensical within this framework.
PREREQUISITESMathematicians, logicians, and students of set theory who are interested in the foundations of mathematics and the properties of ordinal numbers.