Historical Constructions
The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.
When you form the union of two disjoint sets, one would expect the measure of the result to be the sum of the measure of the two sets. A measure with this natural property is called finitely additive. While a finitely additive measure is sufficient for most intuition of area, and is analogous to Riemann integration, it is considered insufficient for probability, because conventional modern treatments of sequences of events or random variables demand countable additivity.
In this respect, the plane is similar to the line; there is a finitely additive measure, extending Lebesgue measure, which is invariant under all isometries. When you increase in dimension the picture gets worse. The Hausdorff paradox and Banach–Tarski paradox show that you can take a three dimensional ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1. Obviously this construction has no meaning in the physical world. In 1989, A. K. Dewdney published a letter from his friend Arlo Lipof in the Computer Recreations column of the Scientific American where he describes an underground operation "in a South American country" of doubling gold balls using the Banach–Tarski paradox. Naturally, this was in the April issue, and "Arlo Lipof" is an anagram of "April Fool".
Read more about this topic: Non-measurable Set
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“Culture is the name for what people are interested in, their thoughts, their models, the books they read and the speeches they hear, their table-talk, gossip, controversies, historical sense and scientific training, the values they appreciate, the quality of life they admire. All communities have a culture. It is the climate of their civilization.”
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