the purpose of induction After getting an answer (in a comment) from peter for this question I have a follow up question. 
If, in all horses are the same color problem for example, we need to use reason, reason which is specific to the case, in order to find that "hole" between the correct base case and the correct inductive step. That "hole" where n=2 which makes the proof collapse.
So how can we verify that our hypotheses are correct and that there are no "holes"?
I mean if we proved the base case p(0) or p(1) or p((int)whatever) is true and the inductive step is true, how can be sure that there no "holes" in the hypothesis?
Do we use induction only to prove what we know, beyond any doubt, is true?
But doesn't that contradicts the purpose of a proof?
I know this is such a Newbie question.. but I have searched for answers for these specific questions and I could not find..
Thank you all!

Actually, the principle of induction is an axiom that is selected to look after that the set of all natural numbers is the smallest possible of all candidates:
$(0\in M\wedge\forall n\in M(n\in M\implies n+1\in M))\implies \mathbb N\subseteq M$.
The axiom is used to prove that certain conditions $p(n)$ is true for all numbers that is elements in the defined set $\mathbb N$, by $M=\{n\in \mathbb N|p(n)\}$.
