I have known the principle of mathematical induction for a long time on set of natural numbers.
Recently, i began reading mathematical logic books and learned about inductive and recursive definition of sets.Now i'm trying to relate the two.
1 - Can we generalize the principle of mathematical induction to prove the property of the element of some general set ( instead of the set of natural numbers ) ?
1 . 1 - If yes, should the general set be able to be inductive ( recursively ) defined so that the principle of mathematical induction will work in prove some property of its elements ?
1 . 1 . 1 - I f this is a necessary condition, there is a way to know if some set can be inductively defined or is it more of a guess ?
My question is really defined in three-layers , the second layer only makes sense if the first layer is true, and the third layer only makes sense if the previous layers are true.
Regardless, of what layers make sense, i'm curious about justification and some extra information ( i know nothing about structural, transfinite and noetherian induction, should i learn those to understand my question ? ) .
P.S.: By inductive definition of a set $E$ i'm meaning this:
We are interested in defining the smallest subset of a fixed set $E$ that includes chosen elements and that is closed under certain operations defined on $E$.That is the lowest base case or lowest subset. The recursive or inductive cases show how we can construct the next level subset, from a current level. In the end, the total set to be defined consists of a sequence of various subsets ( each one representing a level and being indexed by a natural number in the sequence ).