Suppose that $A$ is a finite set. $<_1$ and $<_2$ are two well-orderings on $A$. Suppose that I want to find a formula that repesents the function $F$ that says " send least element in ordering $<_1 $to the least element in the ordering $<_2$ and the next to the next"
What could this formula be?
I think that I can construct such a formula using many "steps". First I define a formula that says "assign least element to least element" and define a formula for every pair of "next elements" and then take all of those formulas and use them to define the required formula!
Another way is to use "self-reference" to construct the formula in which case we are out of First order formulas..
My question is: Is there any simpler way to define it?
Why is such formula important for me?
I need this formula to apply transfinite recursion theorem to find an isomorphism between any two well-orderings on finite set $A$. I can solve the problem without such a heavy formlism but I'm curious about finding such formula.
I believe that you want
$$F(a)=\min_{<_2}\big(A\setminus\{F(x):x\in A\text{ and }x<_1 a\}\big)\;,$$
where for $\varnothing\ne S\subseteq A$, $\min\limits_{<_2}S$ is the least element of $S$ with respect to $<_2$.