A friend who studies pure mathematics ask me to think about the following problem.

Suppose that there is an algorithm named X that has 2 inputs: A and a_1...a_n, where 'A' stands for an arbitary algorithm and 'a_1..a_n' are inputs of A. X receives A and its inputs and returns true if A with a_1..a_n could be terminated, and false if A with a_1..a_n inputs fall into an infinite loop (never ends). Like this:

A(n):
   while(n<5):
      write "I'm immortal!"

and the result of X(A,6) is true and X(A,2) is false.

So what is the result of X(X,X)?

Also, do you know who was the first to introduce this problem?