A miraculous machine
tosses a coin an infinite number of times in one second and the
result is recorded as an infinite sequence of zeros and ones (one
for a head and zero for a tail). Suppose the machine is reset and
the tosses of the coin for the next second are recorded in the
same way. Further suppose that the machine goes on doing this for
ever. The record of the results is an infinite set of infinite
binary sequences.
Show that there is some sequence of tosses that is never recorded
by the machine.
Show that the infinite set of finite (or terminating) binary
sequences can be written as an ordered list whereas the infinite
set of all infinite binary sequences cannot.