What IS the successor function without saying $S(n) = n + 1$? I get really frustrated that all these real analysis books and online webpages say $S(n) = n + 1$ but then say addition is defined in terms off the relationships $a + 0 = a$ and $a + S(b) = S(a+b)$.
I feel like this is a bit of a circular definition because we haven't really defined $S(n)$ and I don't like labeling it "$S(n) = n + 1$" because that feels like a cheap way of appealing to intuition when the whole point of analysis is to rigorously define the very things we normally take for granted and find obvious so that we more accurately understand what we are and are not permitted to do with these numbers.
Can we just treat it as a mapping to some other distinct element? If I look at it this way then I would end up defining axioms like this:


*

*Zero is a number.

*If $a$ is a number, then the successor of $a$, denoted $S(a)$, is a number. 

*$a$ and $S(a)$ are considered distinct numbers.

*If two numbers have the same successors, then they themselves are equal numbers.

*Zero is not the successor of any other number.
So I am envisioning a sort of linked-list relationship:
$$0 \rightarrow \alpha \rightarrow \beta \rightarrow \gamma \rightarrow \delta \rightarrow \epsilon \rightarrow \zeta \rightarrow \eta...$$
The labels are arbitrary but I am seeing natural numbers as just nodes linked together where successor just means "whatever this node points to". So you wouldn't see something like $\alpha$ (or anything else) point to two different nodes, or anything pointing to $0$, or anything pointing to itself.
And then if we want to compute $\beta + \gamma$ using our definition of addition we see that:
$$\beta + \gamma = \beta + S(\beta) = S(\beta + \beta)$$
$$\beta + \beta = \beta + S(\alpha) = S(\beta + \alpha)$$
$$\beta + \alpha = \beta + S(0) = S(\beta + 0)$$
$$\beta + 0 = \beta$$
Combining:
$$\beta + \gamma = S(S(S(\beta)))$$
Which makes sense intuitively, taking the third successor to $\beta$. And if we want to define $\beta$ relative to $0$ or its eventual node in the list we could replace it with successors and then show that $$\beta + \gamma = S(S(S(S(\alpha)))) = S(S(S(S(S(0))))) = \epsilon$$
I mean is this the right way to think of it? Am I right to find $S(n) = n + 1$ problematic or am I missing some point as to why it's always defined this way?
 A: Yep, that works as a successor function! You do need to make sure the number concept is the minimal concept, otherwise you might have a set that has more than the nodes in the linked-list. Otherwise, the way your doing it is actually pretty swell, and you're right to find the "naive successor function definition" problematic and cyclic.
A successor function $S:\mathbb{N}\to \mathbb{N}$ is any function from  which follows the following properties:


*

*For all $x\in \mathbb{N}, S(x) \neq x$

*$S$ is one-to-one.

*There is some element $e\in\mathbb{N}$ such that, for all $x\in \mathbb{N}, S(x) \neq e$

*Also, $\mathbb{N}$ is the minimal set on which you can define such an $S$.


Any such function and any such set can be one type of successor function. Note, the only things needed for that definition is logic, sets, and functions. I never use addition or zero or any sembalence of numbers. Also, the successor function needn't be unique; multiple different implementations can fit this "successor interface." You can check out a video by PBS on defining the successor function here.
A: Your approach is a good one to define the set of whole numbers, that is the set of non-negative integers.
I would add the word whole to your axioms to read as :
$1)$ Zero is a whole number.
$2)$ If $a$ is a whole  number, then the successor of $a$, denoted $S(a)$, is a 
 whole number.
$3)$ $a$ and $S(a)$ are distinct whole numbers.
$4)$ If two numbers have the same successors, then they themselves are equal numbers.
$5)$ Zero is not the successor of any whole number.
Then you may mention  that we denote $S(n)$ by $n+1$ and go from there. 
