I get the set theory and bijection parts, what I don't get is the difference between:

Imagine you have a list that links every integer to every real number between 0 and 1. Also let the real numbers be in binary(that means every number only consist of 0s and 1s. 0.5 would then be 0.1, 0.25 would be 0.01 and so on.) 1 => (0.10100101010...) 2 => (0.10001010101...) 3 => (0.01010101000...) 4 => (0.10000101011...) 5 => (0.00001000010...) Now we can construct a number that differs from every number on this list. Remember, two real numbers are different if at least on digit is different. So let our number be 0.01110.... If we look closely, the number differs from the first number on the list at the first digit, from the second number on the list at the second digit, from the third number on the list at the third digit and so on. So if we construct our number like this, we can be sure that it isn't on the list. So that means that there cannot be a complete list that links every natural number to a real number and every real number to one natural number, because we can always contrust a real number that isn't on the list.