By John Stillwell
While so much texts on genuine research are content material to imagine the genuine numbers, or to regard them purely in short, this article makes a major research of the true quantity method and the problems it brings to mild. research wishes the true numbers to version the road, and to help the recommendations of continuity and degree. yet those possible easy necessities result in deep problems with set theory—uncountability, the axiom of selection, and big cardinals. actually, almost the entire options of countless set conception are wanted for a formal realizing of the genuine numbers, and therefore of research itself.
By targeting the set-theoretic points of study, this article makes the easiest of 2 worlds: it combines a down-to-earth advent to set conception with an exposition of the essence of analysis—the research of endless strategies at the genuine numbers. it truly is meant for senior undergraduates, however it can also be appealing to graduate scholars mathematicians who, earlier, were content material to "assume" the true numbers. Its necessities are calculus and simple mathematics.
Mathematical background is woven into the textual content, explaining how the thoughts of actual quantity and infinity built to satisfy the desires of study from precedent days to the overdue 20th century. This wealthy presentation of background, in addition to a history of proofs, examples, workouts, and explanatory comments, may also help inspire the reader. the fabric coated comprises vintage themes from either set conception and genuine research classes, equivalent to countable and uncountable sets, countable ordinals, the continuum challenge, the Cantor–Schröder–Bernstein theorem, non-stop features, uniform convergence, Zorn's lemma, Borel units, Baire services, Lebesgue degree, and Riemann integrable functions.