In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 24 Feb., 01:19, Virgil <vir...@ligriv.com> wrote: > > > > > There is certainly no meaning of "linear" in English > > > > mathematics that is appropriate. > > > > > Then use German mathematics. There it is. > > > > > f(ax + by) = af(x) + bf(y) > > > > With suitable interpretations for f, a, b, x and y, this would makes f a > > linear function. > > It is not hard to find this interpretation in mathematics. > > > > But if f is to be a mapping between the set of all paths of a Complete > > Infinite Binary Tree and the set of all subsets of |N, which is the only > > sort of mapping under consideration when WM claimed linearity, I defy WM > > to come up with an appropriate definition of a,b,x and y which will make > > such an f a linear mapping. > > > > Two binary strings are treated like two real numbers. In fact they are > nothing but representations of real numbers.
That in no way makes any mapping between the set of all such binary strings and the set of all paths of a CIBT into a LINEAR mapping.
At least until WM has formulated both that set of binary sequences and that COMPLETE INFINITE BINARY TREE as a linear spaces over some field , and then shown that his mapping is a linear mapping between those linear spaces. None of which he has done.
> > Simplest logic. Try to find a set that contains its number if it does > not contain its number. Isn't that simple?
How does that apply to, say, the set of von Neumann natural numbers in ZF?
In the von Neumann model a natural does not ever contains its number, only the numbers of previous naturals.
So in that von Neumann model, WM's above objection fails. --