Re: Cantor and the binary tree

Alan Morgan wrote:

>
> I think he's already claimed that the set of finite integers is not a
> well defined concept (although he punted on exactly why that is) so
> he doesn't have to have an answer to this question. I'm not sure why
> he thinks this is an improvement over the alternatives, but to each his
> own.
>
> Alan
> --
> Defendit numerus

Consider x=x, x-x=0, with careful definition of the free variables in a
system, it's possible to consider rigorously, things along the lines
of:

f(x) = Sum_1^oo x

g(x) = Sum_1^oo -x

f(x)+g(x) = 0

Is that so difficult for you to understand? It's because

f(x)+g(x) = Sum_1^oo x-x = Sum 0 = 0

Then, if you replace x with a constant, then it would still be so, that
0=0, quod erat demonstrandum.

That's trivial, there is basically the umbral calculus, or even the
infinitesimal calculus in the sense of some, that have ready methods
for rigorously deriving true statements from expressions involving the
sums of infinite sequences.

If you're going to reinvent the wheel... perhaps you might be
interested in a consistent theory.

In NSA, nonstandard analysis, the sum over all positive integers n of
1/2^n is exactly equal to one.

About the binary tree, again consider the binary tree to represent an
integer. In considering the Cartesian product of the set {0,1}, 2, for
any finite integer it is an element of 2 x 2 x 2 x ... 2, for some n
many copies of the set, for finite integer n. However, due to the fact
that there are infinitely many finite integers, for no given finite
value of n can the Cartesian product contain any finite integer. Thus,
the tree describing the natural integers, or the integer side of the
radix in the integer modulus expression of binary (decimal, etc.) is
the same as that describing the fractional side, it needs to be an
infinite tree. For a given integer, as there are again no leaf nodes
on the tree, after the highest order of magnitude the rest of the
values are zero.

On the real side, that is only so when the reduced fraction has a
denominator that is a power of two.

It is simple to draw a parallel between the polynomial, with finitely
many monomial summands, and the power series, with potentially
infinitely many non-zero monomial summands, and the infinitely many
finite vs. infinite paths to represent integers vs reals, or for that
matter rationals. Another parallel notion is the analysis of
bitstrings finite, and infinite.

As the radix changes from two to three, again the integer side
represents all integers, and those paths on the real side that
terminate in all zeros are basically multiples of the reciprocal of
some power of three. With the radix being four those paths, which were
represented by those of some power of two as four is a power of two,
represent the same numbers as zero-terminating paths as those where the
radix of two.

In the context of the antidiagonal argument, there is a slight
difference two and three and the other integers greater than three.
The point of dual representation can only be shown to not occur as
simply in a radix that is four or greater, where perhaps none deals
with leading zeros.

The set of all sets is its own powerset. There are some relatively
deep and fundamental issues with regards to unlimited induction that
basically show a variety of things, eg via well-ordering, domain, and
transfer principles, that infinite sets are equivalent. Cantor's
powerset or antidiagonal results, or separately the proof about
complete sets dense in the reals such as the reals, are simple and
accessible, and largely they stand by themselves.

It takes a perhaps deeper understanding of the numbers and sets or
other primary objects that comprise the logical system to countermand
those strictures.

In the physical universe, abstractly the Ding-an-Sich, functions
between physical objects and physical objects are physical objects, as
are those, ad infinitum, so there are infinitely many of them. As are
those, via identity of the totality those sets are equivalent.

The universe is infinite, and infinite sets are equivalent.

Alan: deperdit numerus.

Ross F.

.

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