Monad (nonstandard analysis) explained

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

monad (x) = \{y\in R ^{*} \mid x - y isinfinitesimal \}.

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

Notes and References

1. Book: Goldblatt, Robert . Robert Goldblatt . Lectures on the Hyperreals . Springer . Berlin . 1998 . 0-387-98464-X .
2. Web site: Wood. Carol. Carol Wood. The Infinitesimal Monad - Numberphile . youtube.com. Numberphile. en. video. 4 Sep 2015 . 29 Dec 2022.
3. Book: Keisler, Howard . Howard Jerome Keisler . Foundations of Infinitesimal Calculus . 19 June 2022 . . Madison, Wisconsin, USA . 2 . en . .