October 24th, 2016

Dore: Ogre

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The passage from spaces to locales has at least two additional benefits [23]: the Banach–Tarski “paradoxical” decomposition of the unit ball [2] does not work with locales even though we keep the axiom of choice, and the naive definition of random reals does. Let us briefly explain the latter one. An appealing intuitive definition of a random real x∈[0,1] is that it has no special properties, i.e., that it avoids every null set. This will not do, of course, since a singleton is a null set, and so no particular real can be random. Thus, if we attempt to define the space of random reals as the intersection of all full subsets of [0,1], we are left with an empty space. However, the intersection of all sublocales of [0,1] with full measure is a nontrivial sublocale of [0,1] whose measure is still full. This is the locale of random sequences. It is nonspatial, has no points, it disrupts the traditional notion of space, and it captures our intuitive ideas about randomness in a way that usual spaces cannot.
Dore: Ogre

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Haley, who also served in Afghanistan and now lives in Kempner, Texas, said she slit open the envelope, read the letter and nearly fell out of her chair. "It said I had improperly been given a signing bonus to reenlist and that I had to pay all the money back with interest or I would be in violation of federal law," Haley said. "I freaked out." Now, Haley said, she sends the Pentagon $650 a month. She said this is about a quarter of her family's income, and she's afraid they will lose their home.