Monad (non-standard analysis)

In non-standard analysis, a monad (also called halo^[1]) ^[2] 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

\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.

Notes

↑ Goldblatt, Robert (1998). Lectures on the Hyperreals. Berlin: Springer. ISBN 0-387-98464-X.
↑ NumberPhile

References

H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading

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 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

Scientists	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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Monad (non-standard analysis)

See also

Notes

References