 Well, the piano postulates allow us to find the sum of anything that we want to. They are a little bit tedious to work with, and so we want to consider addition again. And so this leads us to a second definition of addition. And you might ask, well, why do you need a second one? Don't you already have one? And the reason is that the piano postulates, as good as they are, actually apply to any set where we can identify what the next one is. So the problem is that the definition of addition from the piano postulates and all the piano postulates, in fact, all apply to any set of ordinals that have a specific order. So the second definition is based on our concept of cardinality. And so the idea is I'm going to let A and B be disjoint sets, that is to say, these are sets that have nothing in common. I'm going to let A be the cardinality of the set A. I'm going to let B be the cardinality of the set B. And I'm going to define A plus B to be the cardinality of the union of the two sets. Well, let's see how this works. So let's take the problem, prove, using the set definition of addition to plus three equals five. And again, this is a prove statement. We want to use a very specific thing in our prove. And we don't want to know that two plus three equals five. We already know that. The problem is showing how the set definition of addition connects to this statement, whose truth we have no doubt of. And since we want to prove using the set definition of addition, we have to, or we should at least very, to begin with, write down what that set definition is. So the definition of A plus B, according to our set definition, I have two disjoint sets where A is the cardinality of the one and B is the cardinality of the other. A plus B is the cardinality of the union. So now we can take our definition as a framework for the proof by doing what's called instantiation. And what we're going to do is we're just going to substitute in the values that we're working with. The definition of A plus B, well, I want to find two plus three. So I'll substitute those in. I want to compare what I'm trying to prove with our definition. So A is two, B is three, and so I'll substitute these values into my definition. The definition of two plus three will let two be the cardinality of the one, three be the cardinality of the other, two plus three is the cardinality of A union B. Now to complete our proof, we need to find sets A and B and then A union B. So here's a useful guideline. As a general rule, anything that is part of the definition we can find without really having to give comment or explanation for the purposes of our proof. So for example, here the definition refers to the cardinality of A, the cardinality of B, the cardinality of A union B, and because the definition specifically invokes these concepts, this means that we can just find these things without having to comment or explain how we found them once we have A and B. So what I do need to do is to find A and B, which are disjoint sets where two is the cardinality of A, three is the cardinality of B. For example, I might use A being the set A, B and B being the set A, B, C. Well no, I can't use that because remember our sets have to be disjoint and these two sets have elements in common. So these two sets are not disjoint sets. So I can use A being the set A, B and B being the set C, D, E, F. Well no, I can't do that either because I need two to be the cardinality of A, which I'm fine with. Again, once I have this set, the definition invokes the cardinality. I didn't have to comment on how I find that cardinality. I can look at the set and say it has cardinality two, and I can look at this set and say it has cardinality three, except I'd be lying. So this isn't the set I want. I need a set with cardinality three. So how about this set? So I have disjoint sets, check. I have cardinality of A being two, check. Cardinality of B being three, check. And so these are the sets I can use. And so I have two being the cardinality of the set A, union B, well that's the cardinality of the set A, B, C, D, E. And without comment, I can say that the cardinality of this set is five. And I don't have to explain again how I found the cardinality of this set. It's part of the definition. I can just go from cardinality of this set to saying that it's equal to five. And again, the proof of the statement is everything in green. So if I want to make a complete answer to this question, the proof requires that I include everything I have here. And this fits into my definition, my set definition of A plus B.