Monad (nonstandard analysis)

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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.[1][2]

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

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

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" (in en) (video). Numberphile. https://www.youtube.com/watch?v=BBp0bEczCNg. 
  3. Keisler, Howard (19 June 2022) (in en). Foundations of Infinitesimal Calculus. Madison, Wisconsin, USA: University of Wisconsin Press. pp. 2. https://people.math.wisc.edu/~hkeisler/foundations.pdf. Retrieved 16 July 2024.