IJEART

Contemporary constructions of the real number line, via Dedekind cuts or equivalence classes of Cauchy sequences, make no use of infinitesimals. However, using another, lesser known approach, one shows easily that the field of real numbers, in whatever way it is constructed, is distilled from a larger ordered field that includes infinitesimals. In particular,  given any complete, Archimedean, ordered field  and any nonprinicpal ultrafilter U on the set  of natural numbers, one may obtain the complete, non-Archimedean ordered field F =  – where Q is the usual representative of the set of rational numbers in F. Then  may be recovered, up to ordered field isomorphism, by forming the quotient F_fin / I, where F_fin is the set of finite elemnents of F, and I is the ideal of infinitesimals in F. We show in this paper that two natural criteria for deciding whether an element of  is an infiinitesimal are equivalent if and only if the ultrafilter U is a P-point. A P-point is a nonprincipal ultrafilter U with the property that whenever  is a partition of  by sets not belonging to U, there is a  such that  is finite for every  It is known that the existence of P-points is independent of the usual (ZFC) axioms of set theory. This observation shows that undecidable objects and independence results play an essential role in mathematical constructions even as fundamental as the construction of the real line.