nduction can be characterized as a bottom-up method in defining some elements of a set. There is however an Top-down method, saying that the set is the smallest set

closed under the rules.

There monotone and non-monotone induction. So when we deal with more variables we should examine tow cases: monotone and non-monotone.

For more information about “inductive Definitions” see: P.Aczel, Inductive Definitions, Chapter 7, In J. Barwise handbook of Mathematical Logic

Thank you prof Costas Drossos

Dear Prof  Costas Drossos 

I  think that the following file is useful :

http://www.qc.edu.hk/math/Resource/AL/Different%20kinds%20of%20Mathematical%20Induction.pdf

Dear Prof. Ibrahim,

Yes, yhe suggested doc is related. However, does not discuss any generalization to abstract  spaces!