Monad (nonstandard analysis)

In today's world, Monad (nonstandard analysis) has become a topic of utmost importance and interest to a wide range of people. Whether due to its impact on society, its relevance in the scientific field or its influence on economic development, Monad (nonstandard analysis) has managed to capture the attention and debate of experts, professionals and citizens around the world. Throughout history, Monad (nonstandard analysis) has played a crucial role in the evolution of different aspects of human life, and its study and understanding are essential to address the challenges and opportunities that arise today. In this article, we will explore in depth the impact, importance and implications of Monad (nonstandard analysis), analyzing its different facets and offering a global view of its relevance in the contemporary world.

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.^[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 15 December 2025.

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