But if you were to code out such a program it would have to still have an upper bound for the size of the numbers you are testing. My implementation would look like this:

  int i = 4;
  bool sumFound = true;
  while(sumFound)
  {
    sumFound = false;
    for(int p = 0; p < MAX_INT && !sumFound; p++)
    {
      if (isPrime(p))
      {
        for (int q = 0; q < MAX_INT && !sumFound; q++)
        {
          if (isPrime(q))
          {
            if (p + q == i)
            {
              i += 2;
              sumFound = true;
            }	
          }
        }
      }
    }
  }

You could use big integers, but it'd still be bound by whatever the max size is for that. This program would eventually terminate. Could you write an implementation where the limitations of the system don't get in the way of the theoretical mathematical limitation? I guess you could build some system which can accommodate arbitrarily large numbers, but even that would be limited by the computer specs.

I guess what I'm saying is, isn't the implementation of the program the test in itself?