In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 3 Apr., 20:25, Virgil <vir...@ligriv.com> wrote: > > In article > > <3d408b46-a74c-4a88-98ff-43d981d43...@t5g2000vbm.googlegroups.com>, > > > > > > > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 3 Apr., 09:26, Virgil <vir...@ligriv.com> wrote: > > > > In article > > > > <2569eb91-7037-483e-be2c-17fce8394...@j9g2000vbz.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 3 Apr., 00:29, Virgil <vir...@ligriv.com> wrote: > > > > > > > > The point being that removing one object from an infinite set does > > > > > > not > > > > > > diminish the infinite number left in the set > > > > > > > That is a good point. Alas induction holds for every natural number. > > > > > > No! > > > > > Your no is wrong. Induction holds for every natural number. > > > > It does not hold for any natural number as an individual other than when > > in the set of naturals or in some other inductive set. > > It does only hold for the set because it holds for every individual.
The only "individual" for which any inductive proof holds AS AN INDIVIDUAL is the first one, For all the rest of them., they are not individuals alone,m they are also successors of other individuals, and without that transfer from each one to its successor, no induction is possible. > Induction has been applied before there were sets in mathematics and > will be applied after (finished infinite) sets will have been > exorcized.
Which will never happen.
Induction will always depend on "If f(x) then f(next x)" for some definition of "next" which does not end. > > > It is the > > inductiveness of the set, not that it is necessarily a set of natural > > numbers in particular, that justifies induction. > > Can you prove by induction that the sum of the first m elements of > your phantasy sets like 1, 1/2, 1/3, ... is m(m+1)/2? This is provable > using induction.
Not outside of Wolkenmuekenheim!
Consider m = 3
1 + 1/2 + 1/3 = 6/6 + 3/6 + 2/6 = 11/6 but m(m+1)/2 = 3*4/2 = 6 > > > For purposes of induction, one does not need natural numbers, though > > they can be used. > > A gang of matheologians have distorted Peano's writings. Peano in his > original paper writes explicitly: "The sign a + 1 means *the successor > of a*, or *a plus 1*." And later he adds: "2 = 1 + 1, 3 = 2 + 1".
There are all sorts of sets other than sets of naturals that satisfy the Peano axioms and which an inductive argument can be based upon:
One valid form of induction is:
There exists a set of objects, N, and a special object such that: 1. The special object is a member of N. 2. For every object in N there is a different successor object also in N. 3. The special object is not a successor object of any object in N. 4. If successors of two objects in N are the same, then the two original objects are the same. 5. If any set contains The special object and the successor object of every object in N, then that set contains N as a subset. > > Any idea why? Perhaps in your matheology 1/3 = 1/2 + 1? In matheology > 1/3 is greater than 1/2.
Outside of Wolkenmuekenheim, we do not need to have 1 + 1/2 + 1/3 = 6 --