An Example from Description Logic Handbook

Question

I dont understand this example very clearly. The example is taken from Description Logic Handbook.

At the last line of the example, "induction is required, hence such reasoning is not first order". That line completely took me off the guard.

your explanation is much appreicated.

Answer 1

The quote "induction is required, hence such reasoning is not first order" implies that any proof based on induction requires second-order arithmetic. Discussion of why the inductive form cannot be derived from first-order axioms can be found here and here.