But, the paradox extands beyond 'halting.'

I think.

You could just as well say, "There can be no program that can determine, in all cases, if a program prints, 'HELLO WORLD'"

Right?

The function(x) would be true if x produces "HELLO WORD" and false otherwise, where 'x' is the program itself. Then... if (f(x)){print "Goodbye World")... etc., etc.