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1. Cardinality

   Linked via "Vitali set"

   Cardinality of Non-Measurable Sets
   The Axiom of Choice (AC) permits the construction of sets that defy standard Lebesgue measure, such as the Vitali set. The cardinality of such exotic sets is frequently related to $\aleph_1$ or $c$, though their non-measurability suggests they occupy a dimension just beyond the grasp of standard physical description.
   Comparison Table of Initial Cardinalities