Standard part function: Difference between revisions

Revision as of 06:49, 18 February 2020

Revision as of 00:49, 17 December 2020

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[[File:Standard part function with two continua.svg|360px|thumb|right|The standard part function "rounds off" a finite hyperreal to the nearest real number. The "infinitesimal microscope" is used to view an infinitesimal neighborhood of a standard real.]]

Nonstandard analysis deals primarily with the pair <math>\mathbb{R}\subsetsubseteq{}^{\ast}\mathbb{R}</math>, where the [[hyperreal number|hyperreal]]s <math>{}^{\ast}\mathbb{R}</math> are an [[ordered field]] extension of the reals <math>\mathbb{R}</math>, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a [[monad (nonstandard analysis)|monad]], or '''halo''') of hyperreals infinitely close to it. The standard part function associates to a [[Wikt:finite|finite]] [[hyperreal number|hyperreal]] ''x'', the unique standard real number ''x₀'' whichthat is infinitely close to it. The relationship is expressed symbolically by writing

:<math>\,\mathrm{st}(x)=x_0.</math>

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If a hyperreal <math>u</math> is represented by a Cauchy sequence <math>\langle u_n:n\in\mathbb{N} \rangle</math> in the [[ultrapower]] construction, then

:<math>\text{st}(u)=\lim_{n\to\infty}u_n.</math>

More generally, each finite <math>u\in{}^{\ast}\mathbb{R}</math> defines a [[Dedekind cut]] on the subset <math>\mathbb{R}\subsetsubseteq{}^{\ast}\mathbb{R}</math> (via the total order on <math>{}^{\ast}\mathbb{R}</math>) and the corresponding real number is the standard part of ''u''.

==Not internal==

The standard part function "st" is not defined by an [[internal set]]. There are several ways of explaining this. Perhaps the simplest is that its ___domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is <math>\mathbb{R}\subsetsubseteq {}^*\mathbb{R}</math>, which is not internal; in fact every internal set in <math>{}^\ast\mathbb{R}</math> whichthat is a subset of <math>\mathbb{R}</math> is necessarily ''finite'', see (Goldblatt, 1998).

==Applications==