By Sindy Dunbar
Desk of Contents
Chapter 1 - Accuracy Paradox & Apportionment Paradox
Chapter 2 - All Horses are a similar colour & limitless Regress
Chapter three - Drinker Paradox & Lottery Paradox
Chapter four - Paradoxes of fabric Implication
Chapter five - Raven Paradox
Chapter 6 - unforeseen striking Paradox
Chapter 7 - Banach–Tarski Paradox
Chapter eight - beach Paradox & Paradoxical Set
Chapter nine - Gabriel's Horn & lacking sq. Puzzle
Chapter 10 - Smale's Paradox & Hausdorff Paradox
Chapter eleven - Borel–Kolmogorov Paradox & Berkson's Paradox
Chapter 12 - youngster Paradox & Burali-Forti Paradox
Chapter thirteen - Elevator Paradox
Chapter 14 - Godel's Incompleteness Theorems
Chapter 15 - Gambler's Fallacy
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Additional info for Logic & Mathematical Paradoxes
Bk, such that for each i between 1 and k, the sets Ai and Bi are congruent. Now let A be the original ball and B be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball A into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set B, which contains two copies of A. The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed.
There are several ways of producing explicit examples and mathematical visualization: the method of half-way models: these consist of very special homotopies. This is the original method, first done by Shapiro and Phillips via Boy's surface, later refined by many others. A more recent and definitive refinement (1980s) is minimax eversions, which is a variational method, and consist of special homotopies (they are shortest paths with respect to Willmore energy). The original half-way model homotopies were constructed by hand, and worked topologically but weren't minimal.
It can be illustrated by considering the situation of two people who have identical information regarding ravens and black objects, and who have identical estimates of the numbers of ravens and black objects. For concreteness, suppose that there are 100 objects overall, and, according to the information available to the people involved, each object is just as likely to be a non-raven as it is to be a raven, and just as likely to be black as it is to be non-black: and the propositions are independent for different objects a, b and so on.