In nonstandard analysis, a monad or also a halo is the set of points infinitesimally close to a given point.
Given a hyperreal number x in R∗, the monad of x is the set
monad ( x ) = { y ∈ R ∗ ∣ x − y is infinitesimal } . {\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.
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References
Goldblatt, Robert (1998). Lectures on the Hyperreals. Berlin: Springer. ISBN 0-387-98464-X. 0-387-98464-X ↩
Wood, Carol (4 Sep 2015). "The Infinitesimal Monad - Numberphile" (video). youtube.com. Numberphile. Retrieved 29 Dec 2022. /wiki/Carol_Wood ↩
Keisler, Howard (19 June 2022). Foundations of Infinitesimal Calculus (PDF). Madison, Wisconsin, USA: University of Wisconsin Press. p. 2. Retrieved 19 April 2025. /wiki/Howard_Jerome_Keisler ↩