How Real are Real Numbers? Wilfrid Laurier University, Tuesday,
22 September 2009  G. J. Chaitin, IBM Research  Abstract: We
discuss mathematical and physical arguments against continuity
and in favor of discreteness, with particular emphasis on the
ideas of Émile Borel (1871-1956).  I'm not going to give a
tremendously serious talk on mathematics today. Instead I will
try to entertain and stimulate you by showing you some really
weird real numbers.  I'm not trying to undermine what you may
have learned in your mathematics classes. I love the real
numbers. I have nothing against real numbers. There's even a
real number --- Ω --- that has my name on it. [See the chapter
on ``Chaitin's Constant'' in Steven Finch, Mathematical
Constants, Cambridge University Press, 2003.] But as you will
see, there are some strange things going on with the real
numbers.  Let's start by going back to a famous paper by Turing
in 1936. This is Turing's famous 1936 paper in the Proceedings
of the London Mathematical Society; mathema