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On Cauchy's Notion of Infinitesimal

The historical literature of nineteenth century mathematical analysis contains much discussion of Cauchy's conception of the continuum, and of the nature of his infinitesimals in particular. In a contribution to this debate, John Cleave [19 72], following Lakatos [19 78], suggests that Cauchy's 'dynamic' conception of infinitesimals can be accommodated within models of the nonstandard real numbers. Cleave interprets Cauchy's infinitesimals as values of null sequences at infinite integers: an infinitesimal x e *R is accessible if there is a null sequence (sn) (i.e. Sn-+O) such that x = SM for some infinite M. (See Cleave [1972] for the basic definitions). Cleave states (Theorem 5) that there are non-standard models *R in which every infinitesimal is accessible; in his proof he claims that this is the case whenever *R is the ultrapower RN/D, with D a non-principal ultrafilter on N. In this note we show that this claim is correct if and only if D is a special kind of ultrafilter-known as a P-point (defined below). Given the continuum hypothesis (CH) or even Martin's Axiom (MA) P-points can be shown to exist; but not all ultrafilters are P-points. Thus, given CH (or MA), Cleave's Theorem is true, but requires a more careful proof. On the other hand, it is consistent with the axioms of set theory (ZFC) that there are no P-points. These matters are discussed in Jech [1978], pp. 25 7-8. Thus it cannot be shown in ZFC that there are models RN/D in which all infinitesimals are accessible.