||8 credit points
||Veldman, W.H.M. (Radboud Universiteit Nijmegen)
||The aim of the course is to offer an introduction to Brouwer's intuitionistic mathematics. L.E.J. Brouwer (1881-1966), also famous for his results in algebraic topology, (the dimension theorem and the Brouwer fixed point theorem), criticized the logicist approach to the foundations of mathematics in his dissertation (1907), and he attacked the use of the principle of the excluded middle in mathematical arguments (1908). He started to develop mathematics as it should be done according to his own guidelines and obtained his main results in the period 1918-1930. He reconstructed real analysis, measure theory, parts of Cantorian set theory and topology. He not only was careful with the logic of his arguments, but also introduced some new axioms. The discussion on the plausibility of these new axioms is of course of a philosophical nature. The course will be given in Brouwer's spirit, that is, from the conviction that his new approach to mathematics is very important, very natural, and deserves to be taken seriously. Also newer results and developments will be discussed, for instance, the importance of the emergence of the notion of algorithm in 1936 for Brouwer's program, and the revivification of this program by the American analyst E. Bishop in 1963.
Provisional table of contents of the lectures.
(i) There are innitely many prime numbers. There are uncountably many real numbers.
(ii) The difficulty of deciding if two real numbers are equal. Counterexamples in Brouwer's spirit.
(iii) The intermediate value theorem.
(iv) Brouwer's Continuity Principle. Real functions are continuous.
(v) Applications of the Continuity Principle. An infinite set that fails to be Dedekind-infinite. A version of the continuum hypothesis.
(vi) Applications of the Continuity Principle, continued: the Borel Hierarchy Theorem.
(vii) The Fan Theorem. Uniform continuity of functions from [0; 1] to R.
(viii) Some equivalents of the Fan Theorem. The approximate Brouwer Fixed Point Theorem.
(ix) Stumps: inductively generated well-founded trees.
(x) Brouwer's Thesis on Bars in N.
(xi) Applications of Brouwer's Thesis on Bars: The Lusin Separation Theorem.
(xii) Applications of Brouwer's Thesis on Bars: Ramsey's Theorem, the principle of Open Induction on Cantor space.
(xiii) Some equivalents of the principle of Open induction on Cantor space.
(xiv) Measure theory.
(xvi) The problem of the completeness of intuitionistic predicate logic.
||In every session there will be a two hours' lecture. Time will be reserved for the discussion of homework problems. Homework may be handed in and then will be commented.
The examination The examination will consist of one or two written tests (if two, then the first one will be half way the course), and a final oral examination.
The two or three parts of the examination will be of equal value for the determination of the final grade.
||There is no book that guides the contents of the course. It may be helpful to consult Heyting's Introduction to Intuitionistic Mathematics and Bishop-Bridges' book on Constructive Analysis. One also might have a look at Brouwer's Collected Works. Background information may be found in the Brouwer biographies by van Stigt and van Dalen. During the course, lecture notes will be provided.
||A prerequisite for the course is familiarity with (undergraduate) mathematics, in particular real analysis.