 Since we want to be able to do more than simply list the natural numbers, we have to define some operations, and so we define addition as follows. n plus 0 is n, and the successor of n plus m is n plus the successor of m. So, for example, let's find 5 plus 1, and, importantly, let's prove our result. So, remember, if you're not using a definition, you're probably not doing a proof. So, since we're talking about an addition, let's pull in our definition of addition. And since our definition of addition refers to the successors of the natural numbers, we might remember how we defined those successors. So, by definition, 5 plus 1, there's nothing in here that says what happens when you add 1. But we do know that 1 is the successor of 0, and so 5 plus 1 is the same as 5 plus the successor of 0. And we also know that n plus the successor of m is the same as the successor of n plus m. And so that tells us that 5 plus the successor of 0 is the same as the successor of 5 plus 0. But wait, there's more. We know what 5 plus 0 is because that's part of our definition. 5 plus 0 is the same as 5. And so the successor of 5 plus 0 is the same as the successor of 5. And that is the natural number 6. Now, an important idea in mathematics is math ever generalizes. And we can often form a proof by generalizing the specific. So in this specific case, we found that 5 plus 1 was the successor of 5 or 6. And if instead of using 5, we use n, we can go through the same steps. And that gives us a useful little result for any natural number. n plus 1 is the successor of n. What if we do something more complicated? How about let's find 2 plus 3 and again prove our result. So again, we can always write down one side of inequality. So we can write down 2 plus 3 equals. And again, our definition of addition doesn't tell us what happens when we add anything other than 0. But we do know that 3 is the successor of 2. So 2 plus 3 is the same as 2 plus the successor of 2. Our definition does say this is the successor of 2 plus 2. And now lather, rinse, repeat. 2 is the successor of 1. 2 plus the successor of 1 is the same as the successor of 2 plus 1. And at this point, let's take advantage of the fact that we did determine that n plus 1 is the successor of n. And so 2 plus 1 is the successor of 2. That's 3. The successor of 3 is 4. And the successor of 4 is 5.