In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 13 Mrz., 18:40, Frederick Williams <freddywilli...@btinternet.com> > wrote: > > Some mathematicians do reject the > > axiom of choice, but I do not know if any have done so because of > > Banach-Tarski. > > Every mathematician does so! Notwithstanding any intermediate > abracadabra the result is that v = 2V and that is wrong in > mathematics.
The 'number' of points in one sphere equals the 'number' of points in sphere with twice the radius, so that v = 4v, even in ordinary geometry, without any axiom of choice required. > > > > I suspect that a good many, on first hearing of the Banach-Tarski > > paradox, thought 'Wow! How about that! Isn't mathematics fun?' And > > perhaps: 'So what happens if Choice is false? Do any loopy things happen > > in that case?' Meanwhile, note that if set theory is consistent, one > > may safely assume either Choice or its negation. > > ZF is not consistent if this result is correct.
Says someone who claims linearity but cannot prove it.
WM has frequently claimed that a mapping from the set of all infinite binary sequences to the set of paths of a CIBT is a linear mapping. In order to show that such a mapping is a linear mapping, WM must first show that the set of all binary sequences is a vector space and that the set of paths of a CIBT is also a vector space, which he has not done and apparently cannot do, and then show that his mapping satisfies the linearity requirement that f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of a field of scalars and x and y are f(x) and f(y) are vectors in suitable linear spaces.
By the way, WM, what are a, b, ax, by and ax+by when x and y are binary sequences?
If a = 1/3 and x is binary sequence, what is ax ? and if f(x) is a path in a CIBT, what is af(x)?
Until these and a few other issues are settled, WM will still have failed to justify his claim of a LINEAR mapping from the set (but not yet proved to be vector space) of binary sequences to the set (but not yet proved to be vector space) of paths ln a CIBT.
Just another of WM's many wild claims of what goes on in his WMytheology that he cannot back up. --