Math Induction may be applied to definitions as well as proofs of some theorem. It can be used to avoid the use of the triple dots ... . Consider the definition of multiplication of two natural numbers: nw. This can be rephrased as an open statement with variable n:
The product nw is defined for all natural numbers n
Let p(n) be the name of this open statement:
p(n): nw is defined.
1w = w; this defines 1w. Therefore p(1): the product 1w is defined.
2w = 1w + w; 2w is now defined because it is the sum of 1w and w, and they are both defined (as well as addition)
3w = 2w + w; 2w is now defined because it is the sum of 2w and w, and they are both defined (as well as addition)
4w = 3w + w;
....................
(k+1)w = kw + w if p(k) has been defined, then p(k+1) is the sum of two defined terms p(k) and w, and is therefore defined.
[3.x] (Inductive definition of a product of a natural number and a number) Let w be any number. For any natural number n, the product nw is defined as follows:
(a) 1w = w;
(b) if kw has been defined, then (k+1)w = kw + w.
There is a dual argument for powers wn.
[3.x+1] (inductive definition of a number raised to a natural number power) Let w be any number. For any natural number n, the expression wn is defined as follows:
(a) w1 = w;
(b) if wk has been defined, then wk+1 = wk w.