WM has several times claimed that the standard bijection from the set of all binary sequences to the set of all paths of a Complete Infinite Binary Tree is a linear mapping from the set of all binary sequences regarded as a linear space over |R to the set of all paths of a CIBT.
While the mapping is easily shown to be bijective, it fails to be a linear mapping as WM describes it:
> The isomorphism is from |R,+,* to |R,+,*. Only in one case the > elements of |R are written as binary sequences and the other time as > paths of the Binary Tree. Virgil is simply too stupid to understand > that.everal flaws in WM's claim that the identity map on induces a linear map on 2^|N.
WM's flaws in making that claim work include, but are not necessarily limited to:
(1) not all members of |R will have any such binary expansions, only those between 0 and 1, so that not all sums of vectors will "add up" to be vectors within his alleged linear space, and
(2) some reals (the positive binary rationals strictly between 0 and 1) will have two distinct and unequal-as-vectors representations, requiring that some real numbers not be equal to themselves as a vectors, and
(3) WM's method does not provide for the negatives of any of the vectors that he can form, so his "space" does not qualify as a linear space in that way, either.
On the basis of the above problems, and possibly others as well that I have not yet even thought of, I challenge WM's claim to have represented the set |R as the set of all binary sequences, much less to have imbued that set of all binary sequences with the structure of a real vector space or the showed the identity mapping to be a linear mapping from one vector space to another.
Note that while WM's model doe not achieve what he claims for it, there is another model, which a reasonably competent mathematician should be able to find, which does make the mapping into an isomorphism of linear spaces.
It is a shame that someone so obviously of limited ability at mathematics as WM should feel himself so driven to try and correct his betters. --