This book provides a comprehensive exposition of a new way of counting (countable) infinite sets, in particular, numerosities that maintain the ancient Aristotle's principle: "The whole is larger than its parts." It also contains an original introduction to nonstandard analysis. The basic principles are presented in an elementary way by using the ordinary language of mathematics. This is to be contrasted with other introductions, where technical notions from logic are used which are not familiar to most mathematicians.This book is the result of a scientific collaboration among different perspectives of mathematics. The first author is a leading figure in variational and topological methods in nonlinear analysis, and has cultivated a broad range of interests in applications of mathematics; while the second author is a specialist in mathematical logic and in the foundations of nonstandard analysis.