infinitesimal

 $\{x0\}$

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 cut  . Thus there is some model M containing B and some $a\in M$ so that the type of a over B is p.

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

Let K be any ordered ring. Then K contains $\mathbf{N}$. We say $K$ is archimedean if and only if for every $a\in K$ there is some $n\in\mathbf{N}$ so that ${\it a}<{\it n}$. 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 $n\in\mathbf{N}$.

References

• 1 Robinson, A., Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy, New Haven, Conn., 1979.
Title infinitesimal Infinitesimal 2013-03-22 13:22:59 2013-03-22 13:22:59 mps (409) mps (409) 12 mps (409) Definition msc 03H05 msc 06F25 msc 03C64 Hyperreal infinitesimal Archimedean