Why mathematical induction is a deductive process?

Answer 1

"Mathematical induction" is a misleading name.

Ordinarily, "induction" means observing that something is true in all known examples and concluding that it is always true. A famous example is "all swans are white", which was believed true for a long time. Eventually black swans were discovered in Australia.

Mathematical induction is quite different. The principle of mathematical induction says that:

* if some statement S(n) about a number is true for the number 1, and

* the conditional statement S(k) true implies S(k+1) true, for each k

then S(n) is true for all n. (You can start with 0 instead of 1 if appropriate.)

This principle is a theorem of set theory. It can be used in deduction like any other theorem. The principle of definition by mathematical induction (as in the definition of the factorial function) is also a theorem of set theory.

Although it is true that mathematical induction is a theorem of set theory, it is more true in spirit to say that it is built into the foundations of mathematics as a fundamental deductive principle. In set theory the Axiom of Infinity essentially contains the principle of mathematical induction.

My reference for set theory as a foundation for mathematics is the classic text "Naive Set Theory" by Paul Halmos. Warning: This is an advanced book, despite the title. Set theory at this level really only makes sense after several years of college/university mathematics study.