infinitesimal

Let R be a real closed field, for example the reals thought of as astructure in L, the language of ordered rings. Let B be some setof parameters from R. Consider the following set of formulas inL(B):

Then this set of formulas is finitely satisfied, so by compactness isconsistent. In fact this set of formulas extends to a unique type pover B, as it defines a Dedekind cut. Thus there is some model Mcontaining B and some $a\in M$ so that the type of a over B isp.

Any such element will be called B-infinitesimal. Inparticular, suppose $B=\mathrm{\varnothing }$. Then the definable closure ofB is the intersection of the reals with the algebraic numbers.Then a $\mathrm{\varnothing }$-infinitesimal (or simply infinitesimal) isany element of any real closed field that is positive but smaller thanevery real algebraic (positive) number.

As noted above such models exist, by compactness. One can constructthem using ultraproducts; see the entry “Hyperreal (http://planetmath.org/Hyperreal)” for moredetails. This is due toAbraham Robinson, who used such fields to formulate nonstandardanalysis.

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

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