Mathematical induction help, please.

Question

Use the second principle of mathematical induction to show that if f(1) is specified and a rule for finding f(n+1) from the values of f at the first n positive integers is given. Then f(n) is uniquely determined for every positive integer n.

My issue is with the second principle... WHAT IS IT?! how does it differ from the first principle?

Answer 1

At a guess (I think I've heard them called this), the first principle is

if $P(0)$ and $(P(n) \implies P(n+1))$ for all $n$ then $P(n)$ for all $n$

and the second principle is

if $P(0)$ and $((P(r)$ for all $r \leq n) \implies P(n+1))$ for all $n$ then $P(n)$ for all $n$

where $P$ is some proposition depending on the natural numbers.

Answer 2

Perhaps, you 've heard of well ordering principle. You know what is first principle of induction. user 73985 given a correct statement of second principle of induction. Indeed There is a theorem that Well ordered principle, First Principle of Induction and Second principle of Induction are equivalent to each other.

Now, Second Principle of induction is logically same with first principle of induction, so there is no difference between them. But the main "difference" we should call is its application for solving a mathematical problem. For example, you can easily prove fundamental theorem of arithmetic with use of second principle of induction, while you can't easily with first principle of induction(Perhaps you can't).