The definition of the binomial coefficients is here: $$\binom{n+1}{k+1}=\binom{n}{k}+\binom{n}{k+1}$$ $$\binom{0}{0}=\binom{n}{n}=1$$ My aim is to define it in set theory on natural numbers, not on rational numbers like this $\binom{n}{k}=\frac{n!}{k!(n-k)!}$.
let's do it by recursion on $n$.
1) Base: $$n=0 \quad\Longrightarrow\quad C_n=\{\langle 0,1\rangle\}$$ 2) Step: $$C_{n+1}(k)=?$$
Surely I can define it using usual recursion theorem and then restrict it to particular domain.
And here I may use some kind of "finite recursion" on $k$. (where I have both sides) How this principle can be formulated? (the output is function on a finite number). Maybe there are articles/books with some useful recursive principles? (I am also interested in mutual recursion and other recursion principles.)