Let R be a real closed field, for example the reals thought of as a structureMathworldPlanetmath in L, the languagePlanetmathPlanetmath of ordered rings. Let B be some set of parameters from R. Consider the following set of formulasMathworldPlanetmathPlanetmath in L(B):


Then this set of formulas is finitely satisfied, so by compactness is consistent. In fact this set of formulas extends to a unique type p over B, as it defines a Dedekind cutMathworldPlanetmath. Thus there is some model M containing B and some aM so that the type of a over B is p.

Any such element will be called B-infinitesimal. In particular, suppose B=. Then the definable closure of B is the intersectionMathworldPlanetmath of the reals with the algebraic numbersMathworldPlanetmath. Then a -infinitesimalMathworldPlanetmathPlanetmath (or simply infinitesimal) is any element of any real closed field that is positive but smaller than every real algebraic (positive) number.

As noted above such models exist, by compactness. One can construct them using ultraproducts; see the entry “Hyperreal (” for more details. This is due to Abraham Robinson, who used such fields to formulate nonstandard analysisMathworldPlanetmath.

Let K be any ordered ring. Then K contains 𝐍. We say K is archimedean if and only if for every aK there is some n𝐍 so that a<n. Otherwise K is non-archimedean.

Real closed fields with infinitesimal elements are non-archimedean: for any infinitesimal a we have a<1/n and thus 1/a>n for each n𝐍.


  • 1 Robinson, A., Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy, New Haven, Conn., 1979.
Title infinitesimal
Canonical name Infinitesimal
Date of creation 2013-03-22 13:22:59
Last modified on 2013-03-22 13:22:59
Owner mps (409)
Last modified by mps (409)
Numerical id 12
Author mps (409)
Entry type Definition
Classification msc 03H05
Classification msc 06F25
Classification msc 03C64
Related topic Hyperreal
Defines infinitesimal
Defines Archimedean