What's so special about $\mathbb N$?

Question

Through recursion, inversion, extension and all kinds of simple shenanigans the successor function leads to definitions of $+, \cdot, /, x^k, \exp, \sin, \partial, \int$ and other special functions. It seems to have uncanny utility. In particular, this set of functions happens to permit concise description of physics.

I ask, is this merely the result of directions math took in our history, or is there really something profound about $+$? Is $\mathbb N$ hard coded somewhere in the ZF axioms, or in the logic used to describe them? Among the uncountably many other functions, could there be another one, which is not trivially derived form $+$ but could match its utility?

Answer 1

$\mathbb{N}$ is "hard-coded" into logic: logical expressions are strings of symbols, and strings have lengths.

The theories of Peano arithmetic and string processing are essentially the same: with string processing, you can do arithmetic with lookup-tables and pushing symbols around. With Peano arithmetic, you can encode strings as digits of numbers.

An example of how this enters the picture can be seen with recursion. If you have an element $a$ of some set, and a function $f$ on that set, then you can talk about repeatedly applying $f$ to $a$. All of the expressions you can get look like $a, fa, ffa, fffa, \ldots$.

The individual terms can be labelled with natural numbers, counting how many $f$'s appear in the arithmetic expression.

Answer 2

In what sense is multiplication defined in terms of addition? Note, for example, that multiplication can't be defined in the first-order theory of addition.

We only get such a definition of multiplication from addition if we go second-order. But then, if we do go second-order, addition too is definable, in terms of successor.

Now, you might ask in the same spirit as the OP: why does the successor function have such an "uncanny utility" among all the numerical functions, that so many functions (addition, multiplication, exponentiation, superexponentiation, factorial, etc. etc.) can be defined in terms of it, in a second-order framework?

But there is something very odd about that question, as if we can first grasp what is involved in talking of the natural numbers, and then wonder "why is successor special?". That makes no sense. To grasp a structure as a natural number structure just is to grasp it as structured by being generated a successor function from an initial number (there is a zero, a unique next number, a unique next number, etc., without repetitions). The unique role of the successor function together with the unique role of the initial number is what makes the natural numbers the natural numbers (up to isomorphism).