Nonstandard analysis

A framework that extends classical mathematical structures with infinitesimal and infinite numbers. It makes infinitesimal calculus rigorous while preserving theorems about the original structures.

Hyperreal number: An ordered field extending the real numbers with nonzero infinitesimal and infinite elements. Hyperreal numbers provide the number system in which nonstandard calculus takes place.
Real number: An element of the complete ordered field used to represent continuous quantities. The real numbers embed into the hyperreals as the standard values.
Nonstandard calculus: A formulation of calculus using infinitesimals and infinite numbers in a nonstandard extension. It expresses limits and derivatives through infinitesimal changes.
Abraham Robinson: A mathematician who developed nonstandard analysis using methods from mathematical logic. His work established a rigorous framework for infinitesimal analysis in the twentieth century.
Transfer principle: A principle that carries first-order statements about a structure to its nonstandard extension. It lets familiar algebraic and analytic facts apply to hyperreal numbers.
First-order logic: A formal system whose statements quantify over individual objects using predicates and relations. The transfer principle applies to first-order statements in the chosen mathematical language.
Derivative: The instantaneous rate of change of a function, defined through a limit of difference quotients. Its value is the standard part of a hyperreal infinitesimal difference quotient.
Gottfried Wilhelm Leibniz: A seventeenth-century mathematician and philosopher who co-developed calculus and used infinitesimal quantities. His calculus supplied the historical infinitesimal reasoning that nonstandard analysis formalized.
Standard part: The map assigning each finite hyperreal its unique real number at infinitesimal distance. It converts infinitesimally precise hyperreal results into ordinary real values.
Model theory: The mathematical study of formal languages, their interpretations, and the structures satisfying their sentences. Model-theoretic constructions provide a rigorous route to nonstandard extensions.

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