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The programme deals with an upheaval in the foundations of mathematics in the nineteenth century, associated with set theory and the notion of number. It is introduced by Graham Flegg, Reader in Ma...thematics at the Open University, who also describes some of the mathematical arguments. The historical discussion is provided by Professor Clive Kilmister of Kings College, London, Contemporary mathematics is usually built up out of Logic and Set Theory. Set Theory, particularly as applied to the transfinite ordinal and cardinal numbers, is largely the work of Georg Cantor. Separately, Gottlob Frege attempted to base arithmetic on logic, but encountered paradoxes which destroyed the intuitive plausibility of doing so. However, in giving a partial account of the foundations of mathematics, a new mathematical argument (non-constructive existence proof) and new mathematical entities (transfinite numbers) were created. These have consequences even for ordinary numbers.
Module code and title: AM289, History of mathematics AM289; 10 19-09-1976 1976 00:24:30 + Show more... Jean Nunn; Neil Cleminson Graham Flegg; Clive William Kilmister BBC Open University Algebra; Cantor; Number theories; Numbers Stills of Poincare, Kronecker and Hilbert over quotes from each on set theory. Graham Flegg introduces the programme. Clive Kilmister discusses the problems which a lack of an adequate definition of irrational numbers posed for Gauss and Bolzano in their work on the fundamental theorem of algebra. Graham Flegg, with an animation, gives an example of this problem - a polynomial x2 - 2 whose root is square root 2. Kilmister explains, in some detail, Frege's contribution to the proof that natural numbers exist. Graham Flegg, using animations, lists Peano's five axioms for the natural numbers. Clive Kilmister gives a brief biographical background of Georg Cantor and then goes on to describe his work on defining infinite sets and distinguishing between different infinite numbers. Stills of Cantor, St. Petersburg, University of Halle, Weierstrass and frontispiece of Cantor's book. An animation is also used. Graham Flegg, with the aid of animations, goes through one of Cantor's proofs that rational numbers are countable. Kilmister explains how Cantor showed that there are different infinite numbers by demonstrating that real numbers between 0 and 1 were different in number from the natural numbers. Animations are used to illustrate his points. Graham Flegg, using an animation, explains briefly how one can demonstrate that transcendental numbers are not countable. Kilmister sums up the results of Cantor's work. He then goes on to relate the impact of Russell's paradox on Frege who was about to publish volume 2 of his 'Basic Laws of Arithmetic' Graphics, animations and stills of Russell and Frege are used to illustrate points. 6HT/71874 00525_3184 945 no