What are properties, operators, and where do we get numbers from? I am a very confused person. My high school teachers told me division by 0 is undefined because "it just is. The mathematicians just did it that way." So when I found out for real why division by 0 is undefined, I realized that every basic thing I thought I learned couldn't be trusted.

I understand axioms as "something we make up to see what happens," postulates as "things we suspect are true, but haven't proven," and theorems as "things we've proven based on the axioms we've picked." Very tidy, I can dig it.

But I can't find anywhere in several textbooks up to Calc 1 that describes what a "property" is. Is it a particular kind of axiom? Or is it a simple theorem that's close to the axioms? I've found out about indicator functions, but it looks like they only describe properties, without being the properties themselves. How can I prove that $a \times 1 = a$?

Relations, I get them. Mappings from a set to a set, based on rules of any kind. And functions are just a kind of relation where we say each thing in the domain set only gets to have one mapping to the range set. And operators being functions being relations makes sense too, except, where do the operators come from? I'm used to functions being things we make up to study polynomials. But how do we "make up" addition? I can think of any number of algorithms to perform addition, and any number of plain English descriptions of addition, but I couldn't write the right-hand side of $(a, b) = ?$. All the functions I've ever seen are just compositions of addition and the other operators. And why do operators have to be functions, anyway? Couldn't we define division by 0 just by saying division is a relation but not a function? Then it could just return the set of all numbers for $a/0$.

And then numbers. There's an infinity of them, whatever kind you care to pick. How? I figure you could lay down the existence of 1 as an axiom, but how could you construct 2 from that without using addition? Another axiom? Then there'd have to be an infinity of axioms, and that can't be workable.

Yes. I am a very confused person. I hope I picked the right tags.
How many "big" questions... I'll try with a couple of them.



For natural numbers, see Peano axioms : two basic "notions" are assumed :


the existence of an "inatial" numebr : $0$
the existence of a "basic" operation : the successor function $S$.


The first fact is "codified" by the first axiom :


  $0$ is a natural number.


The second fact is established by the second axiom :


  For every natural number $n, S(n)$ is a natural number.


These two "simple" axioms are the rules for the "number game" : start from the beginning and go one step after the other, i.e. counting.

We start from $0$ and apply the successor function $S$ to it, getting a new number : $S(0)$. We call it $1$. Then apply $S$ to $1$, i.e. to $S(0)$ and we get $2=S(S(0))$; and so on...

In order to make this basic machinery to work ad infinitum, we need some further axioms; the third one :


  For all natural numbers $m, n, \ \ m = n$ if and only if $S(m) = S(n)$.


We want that every number has a unique successor.

Then we have :


  For every natural number $n, S(n) \ne 0$. 


This axiom is needed in order to avoid that, after a certain "amount" of numbers, we find a "loop" going back to $0$.

Finally, we have the Induction axiom.

With these axioms, we can define the addition operation.



Regarding the "thorny" question of the division by $0$, the issue is quite simple.

We define subtraction as a "derived" operation starting from addition : if $a + b = c$, we want that $c - b = a$.

Thus, from $2 + 0 = 2$, we "derive" : $2 - 0 = 2$.

The same for division with respect to multiplication; from : $a \times b = c$, we "derive" $c/b=a$.

Unfortunately, we have $a \times 0 = 0$ for any $a$. Thus, what is the "expected value" of $n/0$ ? It must be a number $a$ such that, multiplied by $0$ will return $n$. 

But no $a$, when multiplied by $0$ will gives us back the original $n$.

Thus, we are forced to agree with the awkward fact that : the division by $0$ is undefined. 



A (binary) relation is a way to associate things to other things; the world is plenty of them : "$x$ is father of $y$" defines a relation.

The way mathematics formalizes it is :


  $Father = \{ (x,y) \mid x \ \ \text {is father of} \ \ y \}$.


Functions are relations that satisfy an additional condition, the "functionality" condition : for all $x$ there exists at most one $y$ such that ...

Thus, "father of" is not a function, because a father may have more than one son. The relation "son of" instead, is a function : every son has one father (and not two).

An operation is a function, and thus a relation. We can "describe" the sum as a relation in the following way :


  $Sum = \{ ((n,m),k) \mid k=n+m \}$.


Of course this is not the "recipe" to perform additions; we already have to know how to add $n$ and $m$, but it is a way to "decide", for any triple $n,m,k$ if it satisfy the relation or not, because :


  $((n,m),k) \in Sum$ iff $k=n+m$.


Thus, e.g. $((2,3),5) \in Sum$, while $((1,1),1) \notin Sum$.



In mathematical "parlace", a property is something expressed by a theorem regarding an object or a colelction of objects.

Consider e.g. Euclidean geometry : it is a theory regarding objects (the "geometrical" ones) like points, line, circles, triangles, ...

If we consider the Pythagorean theorem, it states that :


  the sum of the areas of the two squares on the legs of a right triangle equals the area of the square on the hypotenuse.


This theorem states that right triangles have the property that : "the sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse".
But how do we "make up" addition? I can think of any number of algorithms to perform addition, and any number of plain English descriptions of addition, but I couldn't write the right-hand side of $(a, b) = ?$. 


It's complicated.

You can start with Peano's Axioms that define the natural numbers. Here is the version I find most useful. 

We define $\mathbb{N}, S, 0$ such that:


$0\in \mathbb{N}$ 
$S$ is a function mapping $\mathbb{N}$ to itself (the so-called successor function)
$S$ is injective.
For all $x\in \mathbb{N}$, we have $S(x)\ne 0$
For all subsets $P$ of $\mathbb{N}$, if $0\in P$ and for all $x\in P$, we also have $S(x)\in P$, then $P=\mathbb{N}$


Then, using the rules and axioms of logic and set theory, we can actually prove (with some difficulty) the existence of a unique addition function $+$ such that:


For all $x,y\in \mathbb{N}$, we have $x+y\in \mathbb{N}$
For all $x\in \mathbb{N}$, we have $x+1=S(x)$.
For all $x,y\in \mathbb{N}$, we have $x+S(y)=S(x+y)$


Then we can prove (also with some difficulty) the usual algebraic properties of addition: associativity, commutativity and cancelability.

The above development in its full detail would require hundreds of lines of formal proof. For obvious reasons, most authors don't bother with such a detailed development, and just "define" addition as above. Some simply include addition and multiplication in their definition of the natural numbers. So, you will have to forgive your high school teachers if they did not stop the class to explain all this. 

I don't want to discourage your questions. Just be aware that a full explanation of some questions may be "beyond the scope" of the textbook or course.