- 6635 South Hall
The hyper-reals are an ordered field containing the real numbers as well as infinitesimals and infinitely large numbers. They have languished for 50 years, spurned by most professionals. The situation recalls the slow acceptance of other extensions of the concept of number. For example, as late as the 1880's, Kronecker disputed the existence of irrational numbers. Recently hyper-reals have been appearing in many areas of math, in part because they offer conceptual simplification and shorter proofs. See Terence Tao's blog http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analys... We will construct the hyper-reals, and it will become evident that, just from this simple definition, one can deduce most things one wants to know. At the end I might say a few words about doing geometry and topology with the hyper-reals.