My plan here has been for my posts to follow along with what I have been studying. As I've been going, I've reduced things down to the "important" parts, and left other things out. However, I've realized that before I start on the next major topic (functions and sets), I want to go back and cover some things I skipped the first time.
So, starting over at the beginning, we are currently assuming that the natural numbers exist. (I know that in some contexts, we could start with sets and derive the natural numbers, but we aren't going that far here.) We also assume that addition exists, and that it has three properties. These properties are obvious but they are also powerful, and we need to define them in order to use them.
Property 1 is closure. This means that if x is a natural number and y is a natural number, x+y is a natural number. On the one hand, this is obvious. On the other hand, it means, for example, that there is no greatest natural number. Assume there is a greatest natural number, and call it g. Then g+1 is a natural number, because of closure, and g+1 is greater than g, so g is in fact not the greatest natural number. Therefore there cannot be a greatest natural number. This is a result which may seem insignificant, but it defies real-world experience. I expect to have more to say about it in a future post.
Property 2 is commutativity. This just means that x+y=y+x. This is again obvious but useful, and really annoying in mathematical contexts where it isn't true. I'm just saying, don't take it for granted.
Property 3 is associativity. This means that (x+y)+z=x+(y+z), or more generally that in a series of additions, the additions can be performed in any order. Once again, obvious but useful.
These three properties mean that we can derive multiplication as repeated addition, and prove that multiplication of natural numbers has the same properties of closure, commutativity, and associativity. Multiplication of natural numbers also has one property that addition does not have: the existence of an identity element. n×1=n for every natural number n, so 1 is called the multiplicative identity. Also, we have a new property that combines addition and multiplication, the distributive property, which states that x(y+z)=xy+xz. Like the other multiplication properties, we can prove this is true from the definition of addition.
Next time I will look at exponentiation and inequality.