About: Overspill

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In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1. * 1 is an element of A, and 2. * for every element n of A, n + 1 also belongs to A, then A = *N In particular:

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  • In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1. * 1 is an element of A, and 2. * for every element n of A, n + 1 also belongs to A, then A = *N If N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case. The overspill principle has a number of useful consequences: * The set of standard hyperreals is not internal. * The set of bounded hyperreals is not internal. * The set of infinitesimal hyperreals is not internal. In particular: * If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal. * If an internal set contains N it contains an unlimited (infinite) element of *N. (en)
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  • In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1. * 1 is an element of A, and 2. * for every element n of A, n + 1 also belongs to A, then A = *N In particular: (en)
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  • Overspill (en)
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