Monad (nonstandard analysis)

In nonstandard analysis, a monad or also a halo is the set of points infinitesimally close to a given point.^[1][2]

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

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

If x is finite (limited), the unique real number in the monad of x is called the standard part of x.^[3]

References

1. Goldblatt, Robert (1998). Lectures on the Hyperreals. Berlin: Springer. ISBN 0-387-98464-X.
2. Wood, Carol (4 Sep 2015). "The Infinitesimal Monad - Numberphile" (video). youtube.com. Numberphile. Retrieved 29 Dec 2022.
3. Keisler, Howard (19 June 2022). Foundations of Infinitesimal Calculus (PDF). Madison, Wisconsin, USA: University of Wisconsin Press. p. 2. Retrieved 24 January 2025.