Let denote the encoding of TM **M**, and let
denote the encoding of string **u**. It is known that the language

is not Turing-decidable.
From this, however, is doesn't follow that if we index the machine-input
strings to a * paticular* machine **M** the resulting set

is not Turing-decidable. * However*, we * do* know that
is not Turing-decidable, where is a TM accepting
. Our second theorem above follows from this fact. (Make sure
you can verify it.)

Thu Apr 25 14:15:16 EDT 1996