In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 25 Mrz., 23:12, Virgil <vir...@ligriv.com> wrote: > > > > > Lets see WM's statement of the inductive principle. > > > Let P(1) > and let P(x) ==> P(x+1) > > Then P(n) at least for every natural number. > > Proof: For P(2) follows from P(1), P(3) follows from P(2), and so on. > > More is not required.
If proof is not required, or even possible, in any system in which induction, or some equivalent, is not assumed.
So WM gets a failing grade!
One acceptable form of induction is:
There exists a set of objects, N, and a zero object, 0, such that 1. 0 is a member of N. 2. Every member of N has a successor object in N. 3. 0 is not the successor object of any object in N. 4. If the successors of two objects in N are the same, then the two original objects are the same. 5. If a set, S, contains 0 and the successor object of every object in S, then S contains N as a subset. --