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The Banach-Tarski paradox download

The Banach-Tarski paradox by Stan Wagon

The Banach-Tarski paradox



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The Banach-Tarski paradox Stan Wagon ebook
Page: 272
Format: djvu
Publisher: Cambridge University Press
ISBN: 0521457041, 9780521457040


For those who aren't familiar, Banach-Tarski refers to a topological/measure theory paradox. Within the continuum (ℝ) of nonterminating decimals. Logical peculiarities confuse our intuitions about “length”, “density”, “volume”, etc. The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partioning into subsets, replacing a set with a congruent set, and reassembly. Google the Banach-Tarski paradox. There are several variations on it, all of which are equivalent. The simplest one is this: Suppose you have a sphere. This is a really odd Banach-Tarski only looks like a logical paradox, because of our modern physics-inspired views about conservation. Most mathematicians today use choice without reservation, but in Hausdorff, Banach, and Tarski's day, this raised some eyebrows. One consequence is the Banach-Tarski theorem, (1924) or Banach-Tarski Paradox. The Banach-Tarski paradox proves how f#cked up the real numbers are. Like 88% of Randall Munroe's readers, I went straight to wikipedia to figure out what the “Banach-Tarski” reference in the title-text meant (the other 12% already knew). The reason the Banach–Tarski theorem is called a paradox is because it contradicts basic geometric intuition. The extraction in question is that of taking one mathematical sphere out of another mathematical sphere, and both being equal to the original — the paradox that was proved by Banach and Tarski. Today's Spiked Math comic is: Joke Explanation:The joke lies in combining the Banach-Tarski paradox with a miracle that Jesus performed:1. 2) It's possible to cut a sphere into 5 pieces and rearrange the pieces to make two spheres as big as the original.

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