Fundamentals of abstract analysis / Andrew M. Gleason

This is a most unusual book, as will immediately be apparent from the perusal of its table of contents. Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems'' from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied). The motivation behind this selection is explained in the introduction, where the author expresses his belief that "set-theoretic mathematics'' is essentially different from mathematics as a science with "a real content which transcends the inadequacies of our efforts to formalize it'', a distinction which he feels has widened the gap between pure and applied mathematics; his goal is therefore to try to convey to students (around their fourth year in the university) the close relationship between "formalized'' and "real'' mathematics... (MathSciNet)

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