Compactness, infinitesimals, and the reals

One of the most famous philosophical applications of model theory is Robinson’s attempt to salvage infinitesimals. Infinitesimals are quantities whose absolute value is smaller than that of any given positive real number. Robinson used his non-standard analysis to formalize and vindicate the Leibnizian approach to the calculus. Against this, the historian Bos has questioned whether the infinitesimals of Robinson's non-standard analysis have the same structure as those of Leibniz. We offer a response to Bos, by building valuations into Robinson's non-standard analysis. This chapter also introduces some related discussions of independent interest (compactness, instrumentalism, and o-minimality) and contains a proof of The Compactness Theorem and Gödel’s Completeness Theorem.