Defining the Order of the Natural Numbers

Question

Is there a way to give the regular partial order of the natural number directly from their definition through the Infinity Axiom? I have only ever seen the partial order of the natural number to be defined after the Peano axioms are proved, in which case a natural number $n$ is defined to be less than a natural number $m$ if there exists a non-zero natural number $k$ such that $n + k = m$.

Answer 1

The ordering of the naturals is actually much easier to define if we use the set-theoretic interpretation via the axiom of infinity: the axiom of infinity guarantees the existence of the ordinal $\omega$ - AoI gives us an inductive set, and then Separation applied to this set gives us the smallest inductive set, which is $\omega$ - and natural numbers are just elements of $\omega$. The ordinals are linearly (indeed, well) ordered by "$\in$," and restricted to $\omega$ this gives the usual ordering on the naturals.

Of course, we care about more than just the ordering of the naturals. Addition, multiplication, etc. of natural numbers are then defined recursively - this is tedious but not actually hard.