The definition of induction and proving of induction in mathematical logic
Sheng Xu-ming · Journal of Central South University(Social Science) · 2004
Why can the truthfulness of general proposition be proved to be truth in limited process by using of the mathematical induction. That is, why can we to prove all objects in aggregation with some natures through proving one or some objects with these natures in aggregation . The reason is that the aggregation proved by induction must be the inductive aggregation that generated from the definition of induction at first ,it is one of the smallest inductive aggregation like natural number aggregation ,it is close. Namely :if primary elements with some natures in the aggregation, and a function can generate new elements with these natures constantly on the basis of the primary elements, other generative elements with such natures can be generated from the generative elements by using of a generative function. It can be concluded that all the elements in this aggregation possess the natures. The clothing of the inductive aggregation is the principle of mathematical induction. It is a implicative proposition that former proposition is true and latter proposition cannot be false. So inductive proof is deductive proposition through proving truthfulness of former proposition (two process of foundation and induction )to be proved that the later proposition (inductive proposition)is truth certainly.