On 26 Mrz., 21:17, Virgil <vir...@ligriv.com> wrote: > In article > <2dc8b38d-3376-4a6c-89e4-ad4b059d8...@r1g2000yql.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> 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. > > 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.
That is a definition of a sequence, not a proof by induction. It is not even a definition of the natural numbers, because even the ordered set N = (0, pi, pi^2, pi^3, ...) obeys your five points.