I have been studying induction for a few days now and am able to produce some proofs, but have some inquiries as to its theory. First, in strong induction, it seems that different sources disagree on whether base cases are necessary to include within the proofs. Is there any clarification which may be provided for this? My main text says they are not required due to the structure of the inductive step. I do, nonetheless, see strong induction proofs with 1 or more base cases displayed.
Also, what is wrong with the following strong induction "counterexampe" I have come up with as a contradiction to this method?
Show that for all natural numbers x, 0 + x = 0 . [natural numbers = 0, 1, 2,...]
by strong induction, for x = 0, there are no the natural numbers less than 0, so 0 + x = 0 is true since the statement (if, for any x, for all k less than x, P(k) holds, then p(x) holds) is vacuously true. Thus, since it is true for 0, we may use strong induction with x = 1 to show that 0 + 1 = 0, an obvious contradiction.