Is this an adequate definition of the natural numbers?
The term number is synonymous with natural number in this context. The terms front and back are used in order to avoid using numbers first and second to define numbers.
$1$ is a number.
The set $\mathbb{N}$ of numbers is the exactly that set of elements appearing as front elements in the pairs of the succession of numbers defined as follows:
The succession of numbers is the set of ordered pairs of numbers including every number exactly once as a front element, and every number other than 1 exactly once as a back element. If $n$ is a front number then the back number, called the successor of $n,$ is denoted $n^\prime$
[Added to address a flaw pointed out in the answer posted by Greg Martin.]
Every number $n$ uniquely terminates a subset of $\mathbb{N}$ (called the segment $\mathbb{S}_n$), such that $1$ is in $\mathbb{S}_n$ and $\mathbb{S}_n$ includes the successor of each of its members other than $n$.
For every number $n$ we define the segment $\mathbb{S}_n^\prime = \mathbb{S}_n\cup{n^\prime}$
We define $\mathbb{S}_1=\left\{1\right\}$
The proposed definition is intended to use a minimum of concepts. My motivation is that I find definitions using variables and other concepts to beg the question of what those concepts really mean. The variable $n$ was used for brevity, and in the simplest possible way.
This is the definition of the natural numbers I use as the standard:
From Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra
I. $1$ is a number.
II. To every number $a$ there corresponds a unique number $a^{\prime},$ called its successor.
III. If $a^{\prime}=b^{\prime},$ then $a=b$.
IV. $a^{\prime}\ne 1$ for every number $a$.
V. Let $A\left(x\right)$ be a proposition containing the variable $x$. If $A\left(1\right)$ holds and if $A\left(n^{\prime}\right)$ follows from $A\left(n\right)$ for every number $n$, then $A\left(x\right)$ holds for every number $x.$
The first four rules amount to sayin the successor function is a bijection between $\mathbb{N}$ and $\mathbb{N}-\left\{1\right\}$ That is what my long-winded discussion of ordered pairs was intended to accomplish.
As written this is circular, although I suspect it could be rewritten to not be circular. However, note that if any set $S$ satisfies the second part of the definition, then so (for example) does $S \cup \{ (x,y), (y,x) \}$ where $x,y$ are any objects that are not elements of $S$. There are many such extensions of $S$ and so this definition doesn't define the natural numbers uniquely. The missing piece is some sort of induction/well-ordering statement that guarantees that every natural number can be found in a sequence of ordered pairs that contains $1$.