Lebesgue Measure Axioms and the Impossibility of Measuring All Sets
This note is for viewers trying to see precisely which measure-theoretic principles are incompatible with assigning lengths to every subset of (\mathbb{R}).
Vitali Set Construction and Non-Measurable Subsets of the Reals
Abide by Reason explains Vitali’s construction to viewers who know Lebesgue measure and Banach–Tarski in outline, but want to see a concrete example of a non-measurable set on the real line.
Axiom of Choice in Vitali Construction and Representative Selection
This note targets viewers grappling with exactly where the Axiom of Choice enters Vitali’s argument and why the set (V) cannot be constructed explicitly.
Vitali Translates and the Countable Additivity Contradiction
Abide by Reason unpacks the key measure-theoretic step showing that assuming a measure for the Vitali set (V) leads to inconsistency.
Lebesgue vs Countable Sets and Why Rearranging Countable Infinity Keeps Length Zero
Abide by Reason contrasts Vitali’s uncountable pathology with the familiar behavior of countable sets for viewers refining their intuition about measure-zero phenomena.