I recently started an Abstract Algebra course so I may not know all of the 'necessary' vocabulary to fully express my question, but here is my attempt. First I need to provide some background thinking.
When beginning my course, a binary operation was defined as a function $F: S \times S \to S$. When dealing with finite sets, we can just list the binary operation pairings $(x,y)$ with $x,y \in S$ each mapped to some $z \in S$ with no problem, but when dealing with infinite sets we need some sort of rule to calculate all of these pairings in $S \times S \to S$ since while the 'list' of pairings may exist we have no way to understand the properties of some infinitely long list without some tools to analyze that list. As an example, something like $a*b = b \cdot a $ on the set $\mathbb{Z}$ (not commenting on whether this is a group or not etc). What I noticed though is that this just 'references' the multiplication operation, which itself has to be defined. Various definitions exist but the simplest is the grade school repeated addition definition and the one I will be going with for this example.
We can then define $a*b = \sum_{i=1}^{b} a$. But this itself references the 'addition' operation. So now we need to define an addition operation. Again various definitions exist, but for simplicity I will go with the Successor Function since we are working exclusively with $\mathbb{Z}^+$ I am not an expert on peano arithmetic so there may be some errors but the general point should be communicated at least, assuming there are errors. We can define addition a + a as $S^{a}(a) = a + a = 2a = c$. We can define multiplication of $3x $ as $S^{a}(S^{a}(a)) = S^{a}(c) = a + a + a = 3a$ and so on. However, unsurprisingly we notice that we first need a definition for function composition since we 'reference' it with $S^{a}(a)$.
Based on what I am seeing here, it looks like each time we try and define a binary operation we just reference some other binary operation. I am assuming this stops eventually, but where does it stop does 'where it stops' true for all binary operations?
If anything needs to be clarified or corrected, I will do so.
Best Answer
This isn't really a question about abstract algebra per se.
The hierarchy is as follows. We begin by asserting the existence of a set $\mathbb{N}$, together with a distinguished element $0 \in \mathbb{N}$ and function $successor : \mathbb{N} \to \mathbb{N}$, which satisfy the (second-order) Peano Axioms or, equivalently, form a "natural numbers object" (a notion from category theory). This requires an axiom. In structural set theory, the axiom would simply be that there is some $\mathbb{N}, 0, successor$ which is a natural numbers object. If you formulate set theory using ZFC, the axiom in question asserts the existence of a specific set $\omega$; one can then define $successor$ and $0$ and prove that they satisfy the axioms.
Just in case you haven't seen it, when we say that $(\mathbb{N}, 0 \in \mathbb{N}, successor : \mathbb{N} \to \mathbb{N})$ is a natural numbers object, we mean that for all sets $A$, $a \in A$, and $next : A \to A$, there is a unique function $f : \mathbb{N} \to A$ such that $f(0) = a$ and $f \circ successor = next \circ f$ (if you don't like the use of function composition, you could instead say $\forall n \in \mathbb{N} (f(successor(n)) = next(f(n)))$). From this basic property, we can prove more general recursion principles, which we will use to define $+$ and $\cdot$.
Once we have a natural numbers object $(\mathbb{N}, 0, successor)$, we can explicitly define $+ : \mathbb{N}^2 \to \mathbb{N}$ as the unique function satisfying a particular recurrence. We can define $\cdot : \mathbb{N}^2 \to \mathbb{N}$ as the unique function satisfying another recurrence, making reference to the already defined $+$. And of course, $1 = successor(0)$.
Once we have these in hand, we can construct $\mathbb{Z}$ using several different methods and define $0, 1 \in \mathbb{Z}$ in terms of $0, 1 \in \mathbb{N}$. We can also define $+, \cdot : \mathbb{Z}^2 \to \mathbb{Z}$ in terms of $+, \cdot : \mathbb{N}^2 \to \mathbb{N}$. And we can define the unary minus sign $- : \mathbb{Z} \to \mathbb{Z}$. One way of doing this is the so-called “Grothendieck completion” construction.
Once we have this in hand, we can construct $\mathbb{Q}$ using a number of different methods. We then have new constants and functions $0, 1 \in \mathbb{Q}$, $+, \cdot : \mathbb{Q}^2 \to \mathbb{Q}$, and $- : \mathbb{Q} \to \mathbb{Q}$, all defined in terms of the operators of the "same name" on $\mathbb{Z}$. Finally, we have a new operator $-^{-1} : \mathbb{Q} \setminus \{0\} \to \mathbb{Q}$, which we define using the construction of $\mathbb{Q}$. One way of doing this is the “field of fractions” construction.
Finally, from there, we repeat the above step with the reals, using one of a number of explicit constructions to generate a new set $\mathbb{R}$ from the pre-existing set $\mathbb{Q}$. We then define the relevant operators on $\mathbb{R}$ in terms of the existing ones for $\mathbb{Q}$.
Finally, an important thing to note. You write
This is sort of true - if you wish to construct a particular function, you will need to write a rule to specify it. But not all functions have a rule that can be written down. In particular, if I tell you $f : S^2 \to S$, you are not allowed to assume there is some rule that can be written down governing the behaviour of $f$.