Standard part function: Difference between revisions

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{{Short description|Function from the limited hyperreal to the real numbers}}

In [[nonstandard analysis]], the '''standard part function''' is a function from the limited (finite) [[hyperreal number]]s to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal <math>x</math>, the unique real <math>x_0</math> infinitely close to it, i.e. <math>x-x_0</math> is [[infinitesimal]]. As such, it is a mathematical implementation of the historical concept of [[adequality]] introduced by [[Pierre de Fermat]],<ref>{{cite journal |last1=Katz |first1=Karin Usadi |last2=Katz |first2=Mikhail G. |title=A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography |journal=Foundations of Science |date=March 2012 |volume=17 |issue=1 |pages=51–89 |doi=10.1007/s10699-011-9223-1 |url=https://link.springer.com/article/10.1007/s10699-011-9223-1 |postscript=The authors refer to the Fermat-Robinson standard part.|arxiv=1104.0375 }}</ref> as well as [[Leibniz]]'s [[Transcendental law of homogeneity]].

The standard part function was first defined by [[Abraham Robinson]] who used the notation <math>{}^{\circ}x</math> for the standard part of a hyperreal <math>x</math> (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in [[nonstandard analysis]]. The latter theory is a rigorous formalization of calculations with [[infinitesimal]]s. The standard part of ''x'' is sometimes referred to as its '''shadow'''.<ref>{{cite journal |last1=Bascelli |first1=Tiziana |last2=Bottazzi |first2=Emanuele |last3=Herzberg |first3=Frederik |last4=Kanovei |first4=Vladimir |last5=Katz |first5=Karin U. |last6=Katz |first6=Mikhail G. |last7=Nowik |first7=Tahl |last8=Sherry |first8=David |last9=Shnider |first9=Steven |title=Fermat, Leibniz, Euler, and the Gang: The True History of the Concepts of Limit and Shadow |journal=Notices of the American Mathematical Society |date=1 September 2014 |volume=61 |issue=088 |pages=848 |doi=10.1090/noti1149 |url=https://community.ams.org/journals/notices/201408/rnoti-p848.pdf}}</ref>

==Definition==

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==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>\R\subseteq {}^*\R</math>, which is not internal; in fact every internal set in <math>{}^*\R</math> that is a subset of <math>\R</math> is necessarily ''finite''.<ref>{{cite book |last1=Goldblatt |first1=Robert |title=Lectures on the Hyperreals: An Introduction to Nonstandard Analysis |series=Graduate Texts in Mathematics |date=1998 |volume=188 |publisher=Springer |___location=New York |doi=10.1007/978-1-4612-0615-6 |isbn=978-0-387-98464-3 |url=https://link.springer.com/book/10.1007/978-1-4612-0615-6 |language=en}}</ref>

==Applications==