Douglas S. Bridges
Douglas S. Bridges

Lecture notes on intuitionistic logic added 25 November 2022

The following sets of lecture notes are available to anyone who may wish to use them, provided their source is acknowledged.

The Lebesgue Integral via Riesz Sequences

These lectures on the Lebesgue on the real line employ the method of F. Riesz , essentially a special case of the Daniell approach to abstract integration. This avoids the somewhat tedious technical detail about measures that is required in standard, measure-theoretic introductions to the Lebesgue integral, and enables us to reach rapidly the key results on convergence of sequences and series of integrable functions.

The later sections of the notes contain material about the L_{p}(R) of p-power integrable functions on R; a development of the Lebesgue double integral, including Fubini's theorem about the equivalence of double and repeated integrals; and a discussion of topics in advanced differentiation theory, such as Fubini's series theorem and the Lebesgue-integral form of the fundamental theorem of calculus.


Topological and uniform spaces

These notes cover general topological spaces, with additional material (not presented in lectures) on uniform spaces. Ideally, time would have allowed me to include material on completeness and compactness in uniform spaces, but ...


Logic, set theory, and Gödel's incompleteness theorems

These notes were the text for a junior/senior-level course. Chapter 1 deals with propositional and predicate calculus, using the method of tableaux and culminating in the Godel-Henkin completeness theorem, the compactness theorem, and the upwards Lowenheim-Skolem theorem. Chapter 2 presents axiomatic Zermelo-Fraenkel set theory.  Godel's two incompleteness theorems, arguably as significant as any theorems on the twentieth century.




Mathematical economics

These notes were prepared over a number of years, starting as far back as 1989 and continuing to the present, for various courses that applied mathematical analysis to (micro)economics in the spirit of Arrow, Debreu, Mas-Collell and others. These people didn't just produce lots of techniques for solving problems: their primary activity was proving theorems. On the one hand (a favourite expression of economists!), the subject matter has clear economic motivation and interpretation; but on the other, the spirit in which the theory is developed is akin to that of pure mathematics. Thus the current course, an honours-level course entitled Mathematical Economics at the University of Canterbury, is aimed at students who are interested in applications of mathematics---in particular linear and nonlinear analysis---in areas other than the sciences and engineering, as well as those who love mathematics for its own intrinsic beauty.

The topics of the main sections on mathematical economics are as follows: preference and utility; the construction of utility functions from preference relations; Pareto optima and competitive equilibria; demand functions and correspondences; The Brouwer and Kakutani fixed-point theorems; the existence of competitive equilibria. In addition, the lecture notes contain five important appendices that deal with topics in linear and nonlinear analysis that are used in, or arise from, the material in the main sections. These appendices are entitled:  separation theorems; the Stone-Weierstraß theorem; an application of the Michael selection theorem; convex sets and boundary crossing; a simple equilibrium model.

The notes themselves are based on material from a number of sources, including the books of Arrow-Hahn (ref. [2]) and Takayama (ref. [27), and some of the author's own work on preference and utility.

Diagram for the proof of McKenzie's equilibrium theorem:


Lecture notes on intuitionistic logic

These notes were for a short series of lectures given as part of a second-year (sophomore) course on logic. They cover axiomatic intuitionistic logic (propositional and predicate calculus); intuitionistic proof trees; Kripke models. The main source of material for the notes was A Course in Mathematical Logic, by J. L. Bell and M. Machover (North-Holland, 1977).