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?
Well I can't say for sure whether there is a non-circular way to justify induction (Ian essentially claims any justification would be infinite), but certainly your 'proof' is not valid justification. All you have 'established', being as generous as I possibly can, is that given your two conditions we can prove that #1, #2 and #3 are true. I do not accept "etc", so you will have to define it or explain what it means and why I should accept your usage of it.