Monad (nonstandard analysis)

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

monad(x)={y∈ℝ*∣x−y 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" (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 4 January 2026.

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Infinitesimals

History|

* Adequality
* Leibniz's notation
* Integral symbol
* Criticism of non-standard analysis
* The Analyst
* The Method of Mechanical Theorems
* Cavalieri's principle
* Method of indivisibles

Related branches|

* Non-standard analysis
* Non-standard calculus
* Internal set theory
* Synthetic differential geometry
* Smooth infinitesimal analysis
* Constructive non-standard analysis
* Infinitesimal strain theory (physics)

Formalizations|

* Differentials
* Hyperreal numbers
* Dual numbers
* Surreal numbers

Individual concepts|

* Standard part function
* Transfer principle
* Hyperinteger
* Increment theorem
* Monad
* Internal set
* Levi-Civita field
* Hyperfinite set
* Law of Continuity
* Overspill
* Microcontinuity
* Transcendental law of homogeneity

Mathematicians|

* Gottfried Wilhelm Leibniz
* Abraham Robinson
* Pierre de Fermat
* Augustin-Louis Cauchy
* Leonhard Euler

Textbooks|

* Analyse des Infiniment Petits
* Elementary Calculus
* Cours d'Analyse

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