Proofs of a mathematical statement or theorem can have different levels of rigor and I have a question about this.
In the method of mathematical induction, there are statements numbered with 1, 2, 3 etc and we need to prove them. The method says that to prove that these infinitely many statements are correct, two conditions are sufficient: 1) show that statement #1 is correct and 2) show that every statement implies its successor. Now, this means that #1 implies #2, and #2 implies #3, etc etc., and therefore, we proved that the two conditions imply that the infinitely many statements are true.
Now I am wondering: what would a mathematician say about the rigor of this proof? is anything missing in the logic of the proof? could he/she demand more rigor? I'm arising this question because a friend of mine has argued that a more rigorous proof needs to invoke the well ordering principle and related things. So could this stuff be considered more rigorous, or perhaps just completely equivalent?