  There is no proof that black holes contain singularities when they are
generated by real physical bodies. Roger Penrose claimed sixty years ago that
trapped surfaces inevitably lead to light rays of finite affine length
(FALL's). Penrose and Stephen Hawking then asserted that these must end in
actual singularities. When they could not prove this they decreed it to be self
evident. It is shown that there are counterexamples through every point in the
Kerr metric. These are asymptotic to at least one event horizon and do not end
in singularities.
