In PA one can define the ordered pair function in term of naturals, i.e. the ordered pair of naturals is a natural itself. let's label these ordered pairs as natural ordered pairs.
Now working in second order arithmetic (formalized in first order logic language) can we speak of existence of a well ordering on $N$ that is defined in terms of being a set of natural ordered pairs, that satisfy the so and so.. well ordering conditions?
What confuses me is that this well ordering itself would be a subset of $N$, so it would well order itself by the way, so it would be a kind of auto-well ordering relation? Is that possible?
My attempts to solve this question to the positive relied on seeing that in the finite world a well ordering of a finite set $X$ must be at least of a size equal to $|X|-1 + |X|-2 + |X|-3 + ..+|X|-|X|$. So if $X$ is of $\aleph_0$ cardinality then the well ordering relation can be of $\aleph_0$ cardinality. I think a mapping showing this possibility for well ordering subset $R$ of $N$, is for the first element $\alpha$ of $R$ to code $\langle 0,1 \rangle$, then we jump by taking $\alpha + 2^n$ to represent the successive elements of the form $\langle 0, n \rangle$, etc.., now to represent pairs of the form $\langle 1,n \rangle$ for $n > 1$, we start with $\alpha+1$ and jump similarly by $+2^n$, this way we can have enough room to well order all $N$ including $R$ itself! But I'm not sure if this solves it?