Cantor declared himself an outspoken opponent of infinitesimal numbers and, in fact, offered an argument through which he aimed to demonstrate the impossibility of their existence—based precisely on set theory. His conclusion does not follow solely from the axioms of set theory, but rather requires an additional premise—called the “chain thesis”—which describes one of the properties commonly attributed to physical space. Paradoxically, despite such a forceful attempt at refutation, Cantor’s theory provided the very framework needed to resolve, once and for all, the foundational problems of infinitesimal calculus. This chapter develops in detail the ontology of nonstandard calculus, showing the adequacy of nonstandard universes for analysis. It presents the transfer method and explains that it is only applicable to entities called “internal”. The chapter also presents methods for dealing with them.