Your argument still explains the induction step. In the end, P[n] => P[n+1] is just a statement that has “nothing to do with induction” once it is formed. You

Your argument still explains the induction step. In the end, P[n] => P[n+1] is just a statement that has “nothing to do with induction” once it is formed. You have obviously the right intuition about induction, but I think you are confused about its exact nature.