By I. P. Natanson

Volume I covers limitless and element units, measurable units and features, the Lebesgue imperative of a bounded functionality, square-summable services, features of finite diversifications, the Stieltjes essential, totally contiguous capabilities, and the indefinite Lebesgue imperative. quantity II addresses singular integrals, trigonometric sequence, convex services, element units in two-dimensional area, measurable capabilities of numerous variables and their integration, set services and their functions within the thought of integration, transfinite numbers, the Baire class, sure generalizations of the Lebesgue imperative, and a few rules from practical research. Many chapters function demanding exercises.

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To achieve such an understanding, the reader may ﬁnd useful a brief account of certain relevant developments in the history of mathematics and of modern formal logic. The next four sections of this essay are devoted to this survey. II The Problem of Consistency The nineteenth century witnessed a tremendous expansion and intensiﬁcation of mathematical research. Many fundamental problems that had long withstood the best efforts of earlier thinkers were solved; new areas of mathematical study were created; and in various branches of the discipline new foundations were laid, or old ones entirely recast with the help of more precise techniques of analysis.

IV The Systematic Codiﬁcation of Formal Logic There are two more bridges to cross before entering upon Go¨del’s proof itself. We must indicate how and why Whitehead and Russell’s Principia Mathematica came into being; and we must give a short illustration of the formalization of a deductive system—we shall take a fragment of Principia—and explain how its absolute consistency can be established. Ordinarily, even when mathematical proofs conform to accepted standards of professional rigor, they suffer from an important omission.

An alternative to relative proofs of consistency was proposed by Hilbert. He sought to construct “absolute” proofs, by which the consistency of systems could be established without assuming the consistency of some other system. We must brieﬂy explain this approach as a further preparation for understanding Go¨del’s achievement. The ﬁrst step in the construction of an absolute proof, as Hilbert conceived the matter, is the complete formalization of a deductive system. This involves draining the expressions occurring within the system of all meaning: they are to be regarded simply as empty signs.