Practically any serious piece of number theory or combinatorics is a candidate that needs to be looked at since in just about any serious case, there are a myriad of explicit and implicit induction arguments going on. They are truly everywhere. Of course, one knows just what one is stepping off of here metamathematically. Mansucript available with proofs upon request. We use epsilon between integers and sets.
Equality is used only between integers.
The intended interpretation of the set variables is finite sets of integers. However, I prefer the present 17 since it is so much like 16 and their is a pattern throughout the axioms of treating multiplication and addition analogously for many purposes. The point of Theorem 2 is that set parameters can be accomodated in bounded induction. EFA in the following sense. This is the 74th in a series of self contained postings to FOM covering a wide range of topics in f.o.m.