Monad (nonstandard analysis)

{{Short description|Named set of points in nonstandard analysis}}

In nonstandard analysis, a monad or also a halo is the set of points infinitesimally close to a given point.{{cite book | last = Goldblatt | author-link = Robert Goldblatt | first = Robert | title = Lectures on the Hyperreals | publisher = Springer | location = Berlin | year = 1998 | isbn = 0-387-98464-X }}{{cite web|last1=Wood|first1=Carol|author1-link=Carol Wood|title=The Infinitesimal Monad - Numberphile |url=https://www.youtube.com/watch?v=BBp0bEczCNg |website=youtube.com|publisher=Numberphile|language=en|format=video|date=4 Sep 2015 |access-date=29 Dec 2022}}

Given a hyperreal number x in R^∗, the monad of x is the set

:$\backslash text\{monad\}(x)=\backslash \{y\backslash in\; \backslash mathbb\{R\}^*\; \backslash mid\; x-y\; \backslash text\{\; is\; infinitesimal\}\backslash \}.$

If x is finite (limited), the unique real number in the monad of x is called the standard part of x.{{Cite book |last=Keisler |first=Howard |author-link=Howard Jerome Keisler |url=https://people.math.wisc.edu/~hkeisler/foundations.pdf |title=Foundations of Infinitesimal Calculus |date=19 June 2022 |publisher=University of Wisconsin Press |location=Madison, Wisconsin, USA |pages=2 |language=en |access-date={{date}}}}