scheme for n-dimensional induction

Question

In slides: http://www.mathdb.org/notes_download/elementary/algebra/ae_A2.pdf I read the scheme for 2-dimensional induction, but Exists an scheme for n-dimensional induction? Thanks in advance!

Answer 1

According to this slides we can construct $k$-dimensional scheme this way:

Let $S(n_1, n_2, \cdots, n_k)$ denote a statement involving $k$ variables $n_1,n_2, \cdots, n_k$. Suppose:

(i) $S(1,1,\cdots, 1)$ is true.

(ii) $S(1,n_2, \cdots, n_k)$ is true for all positive integers $n_2,\cdots, n_k$.

(iii) If $S(n_1,n_2,\cdots, n_k)$ holds for some positive $n_1,n_2,\cdots, n_k$, then $S(n_1+1,n_2, \cdots, n_k)$ is also true.

Then $S(n_1, n_2, \cdots, n_k)$ is true for all positive integers $n_1,n_2, \cdots, n_k$.

Note that sometimes you can prove (ii) using $(k-1)$-dimensional induction putting $T(n_2,\cdots, n_k)=S(1,n_2,\cdots, n_k)$ and prove that $T$ is true for all positive integers by induction(it's an idea in $2$-dimensional scheme).