The Linearity found that WM claimed but could not find!

[Virgil]

WM claimed to have a bijective linear mapping (isomorphism of vector
spaces, or at least modules) from the set of all infinite binary
sequences to the set of all paths in a Complete Infinite Binary Tree.
But WM required the field of scalars of that mapping to be the real
number field, or some subfield or superfield of the reals, none of which
can be made to work.

WM's claimed methodology has elsewhere been show to be fatally flawed.






But the following shows that what WM has shown himself incapable of
doing can be done by using a certain finite field!

I.e., constructon of a truly linear mapping and vector-space isomorphism
from the set of all infinite binary sequences, B, as a true vector
space to the set of all paths, P, of a Complete Infinite Binary Tree, as
an isomorphic true vector space.

First: A general method for construction of linear spaces over an
arbitrary field (F, +, *, 0, 1) as its field of scalars

Given any field, (F, +, *, 0, 1) and any non-empty set S, one can form
a linear space (vector space), out of the set of all functions from S to
F, here denoted by F^S, with (F, +, *, 0, 1) as its field of scalars,
as follows:
Definition of the VECTOR SUM of two vectors (functions from S to F):
For g and h being any two functions in F^S,
define their vector sum, k - g + h, also in in F^S, by
k(s) = g(s) + h(s) for all s in S.
k = g + h is then the vector sum of g and h.
Defining the SCALAR multiple f a vector VECTOR:
For scalar f in F and vector g in F^S
define that scalar multiple of that vector by h = f*g
where h(s) = f*g(s) for all s in S.
Any f*g thus defined is then a scalar multiple of g.

It is straightforward and trivial to verify that, given the operations
of addtion of two vectors and scalar multiples of as defined above, any
such set F^S has become a true vector space over field F.

Even WM should be able to understand and accept this.

Given the unique finite field, F_2, of characteristic two,
having only the members 0 and 1 required of every field,
and S as the set |N of all natural numbers, the set F^S = F_2 ^ |N,
which, as a set, is the set of all binary sequences,
becomes , with the above construction, automatically
a linear-space/vector-space over the given field of
two elements as its field of scalars,
with scalar 0 times any vector giving the zero vector
and scalar 1 times any vector giving that vector again.

The arithmetic of F_2 is given by
0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1,
0 * 0 = 0 * 1 = 1 * 0 = 0, 1 * 1 = 1.

Thus the set of all binary sequences becomes a linear space
or vector space over the unique field of chasteristic two.

One can also represent each path in a Complete Infinite Binary Tree
by a binary sequence, e.g., with a 0 for a left branch and a 1 for a
right branch, and thus the set of such paths can similarly be formatted
into a vector space with each path a infinite binary sequences.

Then the "identity" bijection on binary sequences,
between these identical vector spaces of binary sequences,
automatically becomes a vector space isomorphism
and bijective linear mapping.

But without using the unique field of two elements as the field of
scalars for both the linear spaces involved,
construction of an HONESTLY linear mapping from the given set of
binary sequences as a linear space to the given set of paths
as a linear space appears highly implausible, at least for someone of
WM's demonstrated lack of mathematical ability.


A pity that WM's mathematical skills are so miniscule,
particularly when his ego is so gargantuan.
--

[fom]

Nice.

I am out of practice. I did not even think of taking the
functions into the field as vectors.

As you know, because of WM's finitism, I had been thinking
what might be done in terms of compactness arguments.

Of course, to do what you have done here, WM would have
had to be HONEST about the many presuppositions concerning
infinity that he makes.

[Ross A. Finlayson]

On Mar 20, 5:15 pm, Virgil <vir...@ligriv.com> wrote:
...

>
> I.e., constructon of a truly linear mapping and vector-space isomorphism
> from the set of all infinite binary sequences, B,  as a true vector
> space to the set of all paths, P, of a Complete Infinite Binary Tree, as
> an isomorphic true vector space.
>

...

>
> A pity that WM's mathematical skills are so miniscule,
> particularly when his ego is so gargantuan.
> --

Blah blah isomorphism ... infinite binary sequences ... Complete
Infinite Binary Tree blah blah blah isomorphic.

Consider a function taking [-1,1] to (-1,1) as so.

f_n(x) = nx, x e [-1,1], n-> oo, ran(f_oo(x)) = (-oo, oo)
g_n(y) = ny, x e (-1, 1), n-> oo, dom(g^-1_oo(g_n(y)) = (-oo, oo)

The properties of those functions is that they're linear.

Then, with j = g^-1 o f (or as composed): dom(j) = [-1,1] and ran(j)
= (-1,1).

With the composition of linear properties then j(x>y) > j(y), and
etcetera.

Then, let h_n(a) = na, n->oo, dom(h(a)) = (-1,1), ran(h(a)) = (-oo,
oo), h is linear as above.

Then a + b is h^-1( h(a) + h(b) ). And, a * b is h^-1( h(a) *
h(b) ).

Here a, b e dom(h) with +, * defined s above, h(b), h(a) e ran(h),
with +, * as in R.

c * (a + b)
= h'( h(c) * h( h'(h(a) + h(b)) ))
= h' ( h(c) * (h(a) + h(b)) )
= h' ( h(c) * h(a) + h(c) * h(b) )
= h' ( h(c) * h(a) ) + h' ( h(c) * h(b) )
= c * a + c * b

Multiplication distributes over addition, and addition and
multiplication are closed as sums and products of h(x) are in (-oo,
oo).

Via symmetry, additive inverses exist and there's an additive
identity, zero.

Then, is there a multiplicative identity and inverse. There is a
mulitplicative inverse of h(a) in [0,oo) in R with the natural
definition of multiplication, and, with the continuous and order-
preserving mapping from [0,1), for any a with 0 < a < 1, there exists
1/h(a) in R+, in ran(h), and: h' ( h(a) * 1/h(a) ) e R[0,1).

Basically: "A definition of field operations on the unit interval."

Regards,

Ross Finlayson

[Ross A. Finlayson]

On Mar 20, 11:58 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:

Simply defined with properties of linear functions and the real
numbers, having scalar multiples of the elements of the field from the
field and a zero vector is quite satisfied.

Then, a consideration is how to intuitively understand the continuum
of real numbers thus that in scaling the unit interval to infinite,
that a point right after zero in the unit equates to the unit in the
scaled result.

This is by no means novel in consideration, for example the other day
are discussed on MathOverflow nilpotent, vis-a-vis, invertible,
infinitesimals, a la the differential and Newton's fluxions.

http://mathoverflow.net/questions/124998/was-the-early-calculus-inconsistent

This would basically have in the field here that the fluxion's
fluxions, are the inverses of the fluxions: in the field.

Or: in the field.

Regards,

Ross Finlayson

[fom]

On 3/21/2013 9:59 PM, Ross A. Finlayson wrote:
> On Mar 20, 11:58 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> wrote:

> Simply defined with properties of linear functions and the real
> numbers, having scalar multiples of the elements of the field from the
> field and a zero vector is quite satisfied.
>
> Then, a consideration is how to intuitively understand the continuum

> of real number thus that in scaling the unit interval to infinite,

> that a point right after zero in the unit equates to the unit in the
> scaled result.
>
> This is by no means novel in consideration, for example the other day
> are discussed on MathOverflow nilpotent, vis-a-vis, invertible,
> infinitesimals, a la the differential and Newton's fluxions.
>
> http://mathoverflow.net/questions/124998/was-the-early-calculus-inconsistent
>
> This would basically have in the field here that the fluxion's

> fluxions, were the inverses of the fluxions: in the field.
>

I made mention of Berkeley the other day and compared
his criticism as saying there was an accounting error
of equal proportion on each side of the ledger. (I may
have actually attributed the example to him in error)

In a part of "The Analyst" not shown in the link, he
writes:

"Therefore the two errors being equal and contrary
destroy each other; the first error of defect being
corrected by a second error of excess."

[JT]

Could you give examples for what you and WM try to accomplish in
layman terms, if that is even possible?

[Ross A. Finlayson]

JT, Wolfgang Muckenheim comes across as a finitist. Basically that
means he thinks everything is finite. Virgil Hancher is a dogmatist,
basically he's a self-appointed defender of the orthodoxy, satisfying
his impulse to belittle others with the argument from authority of
modern mathematics. Neither claims much of accomplishing anything,
where "to accomplish" is to develop novel mathematics. Wolfgang wants
to roll it back, Hancher not to roll it forward, basically in the
Bourbaki sense they're negative or strictly negative on the scale of
"accomplish".

Then, here, a definition of field operations for the unit interval is
a bit different than either of those tacks. It's often said that an
interesting property of the complete ordered field is that it's unique
up to isomorphism, as the complete ordered field. It's shown that via
simplest properties of linear functions, a non-real linear function
exists (not a real function, not in some senses of applicability in
the solutions of differential equations linear, but having properties
of continuous and monotone mappings), this function that is invertible
then sees the operations of the field (from abstract algebra, see
properties of fields or field axioms) defined for the interval, from
the reals, (-1,1). With this example then any interval of the reals
centered on the origin has field operations, so equipped with those
operations, each is a field.

And, each is a vector space.

As well then are simple considerations as the unit disc, and Argand or
complex plane, as fields and isomorphic to the complete ordered field.

And, that would be news to some.

Then, what that establishes, is a notion of scaling, of scale, from
the unit, to the infinite, and infinitesimal. For this you might look
to Yaroslav Sergeyev's simple development of powers of infinity as of
scales, the "Infinitarcalcul" of Paul du Bois-Reymond, or even a
variety of developments as seen here.

It might be much different if transfinite cardinals (where powers of
transfinite ordinals are as to scales, but none of their countable
powers reaches the uncountable that modern mathematics has for the
simple continuum), if transfinite cardinals _explained_ something in
physics or were used in continuum analysis for standard results. But,
they aren't. Then, when there's a simple intuition that scaling the
natural integer to fit within the unit, _makes_ a continuous unit
reals, and following up on that intuition: the would-be function
_doesn't_ see the result of uncountability follow, then it's of
interest to a conscientious mathematician: why that is so.

Regards,

Ross Finlayson